Co-ordinate Geometry — practice questions

326 curated practice questions from Co-ordinate Geometry (Mathematics JEE Main, NTA Exams) — 239 easy, 37 hard, 50 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• The set of all real values of for which exactly two common tangents can be drawn to the circles x 2 + y 2 - 4x - 4y + 6…
• The equation of the tangent to the parabola y 2 = 4ax at point (a/t 2 , 2a/t) is
• If the parabolas y^ 2 =4 b(x-c) and y^ 2 = tax have a common normal, then which one of the following is a valid choice…
• The incentre of the triangle with vertices (1, 3 ) (0,0) and (2,0) is,
• In a ∆ABC, if A is the point (1, 2) and equations of the median through B and C are respectively x + y = 5 and x = 4…
• The lines x + y = |a| and ax – y = 1 intersect each other in the first quadrant. Then, the set of all possible values…
• If the point (-5+ 2 ,-3+ 2 ) is an interior point of the larger segement of the circles x 2 + y 2 = 16 cut off by the…
• Equation of tangents to the ellipse x^ 2 9 + y^ 2 4 =1, which are perpendicular to the line 3x + 4y = 7, are
• The eccentricity of the ellipse 25 x^ 2 +16 y^ 2 -150 x-175=0 is,
• The eccentricity of the conic represented by x 2 – y 2 – 4x + 4y + 16 = 0 is
• The equation of a circle which passes through the three points (3, 0), (1, –6), (4, –1) is-
• A straight line L is drawn perpendicular to the line 5x–y=1. The area of the triangle formed by the line L and…
• A variable circle passes through the fixed point A (p, q) and touches x - axis. The locus of the other end of the…
• The circle C has radius 1 and touches the line L at P. The point X lies on C and Y is the foot of the perpendicular…
• If e and e F are the eccentricities of a hyperbola and its conjugate, then the value of 1 e ^ 2 + 1 e ^ 1^ 2 is
• The tangent to the parabola y 2 = 16x, which is perpendicular to a line y – 3x – 1 = 0 is
• Four points (0, -1), (6, 7), (-2, 3) and (8, 3) are the vertices of a
• Equation of the circle which passes through the centre of the circle x 2 + y 2 + 8x + 10y – 7 = 0 and is concentric…
• The equation of common tangent to the curves y^ 2 =16 x and x y=-4 , is:
• If the distance between the foci and the distance between the directrices of the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 =1 are…
• Coordinates of the focus of the parabola x 2 – 4x – 8y – 4 = 0 are
• The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3 x+4 y=9 and…
• The focus of the parabola (x+y)^ 2 =4(x-y-z) is
• Let P be the point of intersection of the common tangents to the parabola y^ 2 =17 x and the hyperbola 8 x^ 2 -y^ 2 =8…
• The angle between the lines represented by the equation x^ 2 -2 p x y+y^ 2 =0 is,
• If the line a x+b y+c=0 is normal to the curve x y=1 , then
• The locus of the vertex of the family of parabolas y = a^ 3 x^ 2 3 + a^ 2 x 2 – 2a is
• The point (–1, 1) and (1, –1) are symmetrical about the line
• The values of m for which the equation 2 x^ 2 +m x y+3 y^ 2 -5 y-2=0 can be expressed as the product of two linear…
• If and are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse…
• The equation of the base of an equilateral triangle is x+y=2 and the vertex is (2, –1). Length of its side is then 10…
• An ellipse has the point (1_ 1 -1 ) and (2,-1) as its foci and x+y=5 as one of its tangent then the coordinates of the…
• The coordinates of an end-point of the latus-rectum of the parabola (y–1) 2 = 4(x+1) are
• If orthocentre and circumcentre of a triangle are respectively (1, 1) and (3, 2), then the coordinates of its centroid…
• The equations of the tangents to the hyperbola x 2 – 4y 2 = 36 which are perpendicular to the line x – y + 4 = 0 are
• The eccentricity of an ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 whose latus rectum is half of its major axis is,
• A point which divides the joint of (1, 2) and (3, 4) externally in the ratio 1 : 1
• Let P be the point (1, 0) and Q a point on the parabola y^ 2 =8 x . The locus of midpoint of PQ is,
• An ellipse has its centre at (1, –1) and semi major axis equal to 8. If this ellipse passes through the point (1, 3)…
• The length of the minor axis (along y-axis) of an ellipse in the standard form is 4 3 . If this ellipse touches the…
• If the line y=m x+7 3 is normal to the hyperbola x^ 2 24 - y^ 2 18 =1 , then a value of m is:
• A circle of radius r is concentric with an ellipse x ^ 2 a ^ 2 + y ^ 2 ~b ^ 2 =1 . If common tangent is inclined to the…
• The abscissas and ordinates of the points A and B are the roots of the equations x^ 2 +2 a x+b=0 and x^ 2 +2 c x+d=0…
• Three straight lines 2 x+11 y-5=0 24 x+7 y-20=0 and 4 x-3 y-2=0 ,
• A square of side a lies above the x-axis and has one vertex at the origin. The side passing through the origin makes an…
• For the family of straight lines bx + ay = ab, which one is not correct.
• The value of λ, for which the circle x^ 2 +y^ 2 +2 x+6 y+1=0 intersects the circle x^ 2 +y^ 2 +4 x+2 y=0 orthogonally…
• If the lines x + 2ay + a = 0, x + 3by + b = 0 and x + 4cy + c = 0 are concurrent, then a, b and c are in
• The equation of ellipse whose axes are coincident with the coordinate axes and which touches the straight lines 3 x-2…
• Let the equations of two sides of a triangle be 3 x-2 y+6=0 and 4 x+5 y-20=0 . If the orthocentre of this triangle is…
• The equation of a tangent to the parabola y^ 2 =8 x is y=x+z . The point on this line from which the other tangent to…
• The equation of the line which passes through the point (1, –2) and cuts off equal intercept from the axes is
• The equation of the tangents to the ellipse 4x 2 + 3y 2 = 5 which are parallel to the line y = 3x + 7 are
• Let P ^ P ( a sec , b ) and Q (a _ 1 b _ 1 ) , where + = 2 . be two points on the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 =1 …
• The equations of the lines through (–1, –1) and making angles 45° with the line x + y = 0 are
• The name of the curve described parametrically by the equations x = t 2 + t + 1, y = t 2 – t + 1 is
• The equation of the circle with centre (2, 1) and touching the line 3 x+4 y=5 is,
• Let a, b, c and d be non-zero numbers. If the point of intersection of the lines 4ax + 2ay + c = 0 and 5bx + 2by + d =…
• The equation of a line through the point of intersection of the lines x – 3y + 1 = 0 and 2x + 5y – 9 = 0 and whose…
• Find the equation of tangents to the ellipse 4 x^ 2 +3 y^ 2 =5 , which are parallel to the line 3 x-y+7=0 .
• Given below are two statements. STATEMENT -1 If the vertices of a triangle are having rational coordination then its…
• If the line y =x +3 meets the circle x 2 + y 2 = a 2 at A and B, then the equation of the circle having AB as a…
• The foci of a hyperbola coincide with the foci of the ellipse x 2 /25 + y 2 /9 = 1 . If eccentricity of the hyperbola…
• ABC is a variable triangle with the fixed vertex C(1,2) and A, B having the coordinates ( t_ 1 t ), ( t_ 1 - t )…
• The equation of the parabola whose vertex is (-1, -2), axis is vertical and which passes through the point (3, 6), is
• The equation of circle with centre (1, 2) and tangent x + y – 5 = 0 is
• Which of the following statements is true regarding the following two circles ? x 2 + y 2 – 4x – 8y = 0 and x 2 + y 2 –…
• The equation of two equal sides of an isosceles triangle are 7x – y + 3 = 0 and x + y –3 = 0 and its third side passes…
• The image of the point (2, 1) with respect to the line mirror be (5, 2). Then the equation of the mirror is -
• The equations of common tangents to y 2 = 4ax and (x + a) 2 + y 2 = a 2 are
• The distance of a point on ellipse x^ 2 6 + y^ 2 2 =1 from its centre is 2. The eccentric angle of the point will be
• If the vertices P, Q, R of a ∆PQR are rational points, which of the following point(s) of the ∆ PQR is not always…
• If x + 2y = 3 is a line and A (–1, 3); B (2, –3); C (4, 9) are three points, then :
• Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of chord of the…
• The locus of the point of intersection of tangents to the parabola y^ 2 =4(x+1) and y^ 2 =8(x+2) which are…
• If e 1 and e 2 are the eccentricities of the ellipse, x^ 2 18 + y^ 2 4 =1 and the hyperbola, x^ 2 9 - y^ 2 4 =1…
• The abscissae and ordinates of the points A and B are the roots of the equations x 2 + 2ax + b = 0 and x 2 + 2cx + d =…
• Assuming that the line x – 3y + 4 = 0 is working as a mirror for the point (1, 2) then the coordinates of the image is
• The equation of the tangents to the circle x 2 + y 2 = 4, which are parallel to x + 2y + 3 = 0, are
• Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity 3 .
• The line y = 2t 2 intersects the ellipse x^ 2 9 + y^ 2 4 =1 in real points, if
• Let a and b be non-zero real numbers. Then, the equation (a x^ 2 +b y^ 2 +c ) (x^ 2 -5 x y+6 y^ 2 )=0 represents,
• The ratio in which the line x + y = 4 divides the line joining the points (–1, 1) and (5, 7) is 1: α then α =
• The circle x 2 + y 2 – 2x – 3ky – 2 = 0 passes through the fixed point :
• If P ≡ (1, 0), Q ≡ (–1, 0) and R ≡ (2, 0) be three given points, then the locus of the point S satisfying the relation…
• The number of points outside the ellipse on its major axis from which a normal (other than the x-axis) can be drawn to…
• Vertex, focus, latus rectum, length of the latus rectum and equation of directrix of the parabola y 2 = 4x + 4y are
• The equation of the incircle of the triangle formed by the axes and the line 4x + 3y = 6 is
• The equation of the circle passing through (1, –3) & the points common to the two circle, x 2 + y 2 – 6x + 8y – 16 = 0…
• The length of the perpendicular from the origin to a line is 7 and the line makes an angle of 150° with the positive…
• The equation of the circle described on the chord 3x + y + 5 = 0 of the circle x 2 + y 2 = 16 as diameter is
• Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y), passing through origin…
• The parametric representation of a point on the ellipse whose foci are (–1,0) and (7, 0) and eccentricity 1 2 is
• The circle x 2 + y 2 – 3x – 4y + 2 = 0 cuts x-axis at
• Let P be the point (1, 0) and Q a point on the locus y 2 = 8x. The locus of mid point of PQ is
• The equation of the circle passing through (1, 1) and the points of intersection of x^ 2 +y^ 2 +13 x-3 y=0 and 2 x^ 2…
• If a sphere of constant radius 'k' passes through the origin and meets the axis in A, B, C then the centroid of the…
• The locus of the equation x 2 – 5x + 6 = 0 in 2D, is
• Two vertices of an equilateral triangle are (–1, 0) and (1, 0), and its third vertex lies above the x-axis. Then the…
• Let s= (x, y) R^ 2 : y^ 2 1+r - x^ 2 1-r =1 ; r 1 . Then S represents
• The line joining two points A (2, 0); B (3, 1) is rotated about A in the anticlockwise direction through an angle of 15…
• The equation of the circle which touches the axis of y at the origin and passes through (3, 4) is-
• The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2…
• If two circles x^ 2 +y^ 2 =r^ 2 and x^ 2 +y^ 2 -10 x+16=0 intersect at two real points, then
• The length of the latus rectum of an ellipse is one third of the major axis, its eccentricity would be
• The equation λx 2 + 4xy + y 2 + λx + 3y + 2 = 0 represents a parabola, if λ is
• If a circle passes though the point (a, b) and cuts the circle x^ 2 +y^ 2 =4 orthogonally, then the locus of its centre…
• The co-ordinate of the point on the circle x 2 + y 2 – 12x – 4y + 30 = 0, which is farthest from the origin are:
• The point of contact of the line 2 x+ 6 y=2 and the hyperbola x^ 2 -2 y^ 2 =4 is,
• If the vertices of a triangle are [a t_ 1 t_ 2 , a (t_ 1 +t_ 2 ) ], [a t_ 2 t_ 3 , a (t_ 2 +t_ 3 ) ], [a t_ 3 t_ 1 , a…
• A ray of light along x+ 3 y= 3 gets reflected upon reaching x-axis, the equation of the reflected ray is
• The name of the conic represented by the equation x 2 + y 2 – 2xy + 20x + 10 = 0 is
• If the lines 3 x-4 y-7=0 and 2 x-3 y-5=0 are two diameters of a circle of area 49 square unit, then equation of the…
• The equation of the parabola whose axis is parallel to y-axis and which passes through the points (0, 4), (1, 9) and…
• The locus of a point, whose abscissa and ordinate are always equal is
• The circle x^ 2 +y^ 2 =4 x+8 x+5 intersects the line 3 x-4 y=m at two distinct points if
• A point on the parabola y 2 = 18x at which the ordinate increases at twice the rate of the abscissa is
• The equation of the circle touches y axis and having centre (–2, –3) is
• The point P is equidistant from A(1, 3), B(- 3, 5) and C(5, - 1), then PA is equal to
• The line passing through (0, 1) and perpendicular to the line x – 2y + 11 = 0 is
• The minimum distance of any point on the ellipse x^ 2 +3 y^ 2 +2 x y=4 from its centre is
• Equation of a circle concentric with the circle 2x 2 + 2y 2 – 6x + 8y + 1 = 0 and of double its area is
• The number of common tangents to the circles x^ 2 +y^ 2 =4 and x^ 2 +y^ 2 -6 x-8 y=24 is,
• The equations of the tangents to the circle x 2 + y 2 – 6x + 4y – 12 = 0 which are parallel to the line 4x + 3y + 5 =…
• The eccentricity of an ellipse, with its centre at the origin is 1 2 . If one of the directrix is x=4 , then the…
• The line passing through (1, 1) and parallel to the line 2x – 3y + 5 = 0
• If x_ 1 , x_ 2 , x_ 3 as well as V_ 1 , V_ 2 , V_ 3 are in GP with same common ratio, then the points P^ P (x_ 1 , Y_ 1…
• If from any point P on the circle x 2 + y 2 + 2gx + 2fy + c = 0 tangents are drawn to the circle x 2 +y 2 +2gx +2fy +…
• The circles x 2 + y 2 + 6x + 6y = 0 and x 2 + y 2 – 12x – 12y =0
• The minimum area of a triangle formed by any tangent to the ellipse x^ 2 16 + y^ 2 81 =1 and the co-ordinate axes is:
• The equation of the latus rectum of the parabola x 2 = –12y is–
• The equation of the common tangent touching the circle (x-3)^ 2 +y^ 2 =9 and the parabola ^ 2 = 4 above the x-axis is,
• The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x = 4, then the…
• If H ( x , y )=0 represents the equation of a hyperbola and A(x, y)=0, C(x, y)=0 , the equations of its asymptotes and…
• The equation of a tangent to the circle x 2 + y 2 = 25 passing through (–2, 11) is
• The locus of the centre of a circle, which touches externally the circle x^ 2 +y^ 2 -6 x-6 y+14=0 and also touches the…
• The parametric equation of a parabola is x = t 2 + 1, y = 2t + 1 . The Cartesian equation of its directrix is
• The line 3x – 2y = k meets the circle x 2 + y 2 = 4r 2 at only one point, if k 2 is
• The equation | r |^ 2 - r [2 i +4 j -2 k ]-10=0 represents a
• If the two circles (x – 1) 2 + (y – 3) 2 = r 2 and x 2 + y 2 – 8x + 2y + 8 = 0 intersect in two distinct points, then
• The equation of the latus rectum of the ellipse 9x 2 + 4y 2 –18x – 8y – 23 = 0 are
• The locus of the point of intersection of the tangents to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 if the difference of the…
• Two common tangents to the circle x^ 2 +y^ 2 =2 a^ 2 and parabola y^ 2 =8 a x are,
• If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent…
• The equation of sphere, concentric with the sphere 2 x^ 2 +2 y^ 2 +2 z^ 2 -6 x+2 y-4 z=1 and double its radius, is
• If focus of a parabola is (2, 0) and one extremity of latus rectum is (2, 2), then its equation is
• The locus of a point which divides the line segment joining the point (0, -1) and a point on the parabola, x^ 2 =4 y …
• If the normal at any point P on the ellipse cuts the major and minor axes in 'G' and 'g' respectively and 'C' is the…
• The maximum distance from origin of a point on the curve x=a t-b ( a t b ) and y= t-b ( a t b ) . both a, b > 0 is,
• If the line 2 x+3 y=1 touches the parabola ^ 2 = 4 a at the point P , then find the focal distance of the point P.
• The equation of the parabola whose focus is (–1, 1) and directrix is 4x + 3y – 24 = 0 is
• The circles x 2 + y 2 + 2x – 2y + 1 = 0 and x 2 + y 2 – 2x – 2y + 1 = 0 touch each other
• Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3,1) and has…
• The locus of the vertices of the family of parabola y= a^ 3 x^ 2 3 + a^ 2 x 2 -2 a is,
• A line of fixed length a + b moves so that its ends are always on two fixed perpendicular straight lines; then the…
• If the vertices of ΔABC are A(5,0), B(3,4) and C[ 5 , 2 5 ) , then the coordinates of the orthocentre is
• If latus rectum of the hyperbola is half of its transverse axis, then its eccentricity is
• A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is
• The values of 'm' for which a line with slope m is common tangent to the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 =1(a b) and…
• The equation of director circle of the hyperbola x^ 2 16 - y^ 2 4 =1 is given by,
• If the two circles x^ 2 +y^ 2 +2 a x+c=0 and x^ 2 +y^ 2 +2 b y+c=0 are touching each other, then
• The co-ordinates of the foot of perpendicular from (a, 0) on the line y = mx + a ~m are
• The equation of the sphere with centre (3, 6, -4) and touching the plane 2 x-2 y-z-10=0 is :
• The centres of those circles which touch the circle, x^ 2 +y^ 2 -E x-E y-4=0 , externally and also touch the x-axis…
• The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept – 4 . Then, a possible value…
• If O is the origin and Q is a variable point on y 2 = x. Find the locus of the mid-point of OQ.
• Three distinct points A, B and C are given in the 2-dimensional coordinate plane such that the ratio of the distance of…
• The length of the chord of the parabola x^ 2 =4 y having equation x- 2 y+4 2 =0 is:
• The two conics bx 2 = y and x^ 2 a^ 2 - y^ 2 b^ 2 =1 intersect iff -
• If the equation of the tangent to the circle x 2 + y 2 – 2x + 6y – 6 = 0 parallel to 3x – 4y + 7 = 0 is 3x – 4y + k =…
• Let P(4, -4) and Q(9, 6) be two points on the parabola, y 2 = 4x and let x be any point on the arc POQ of this…
• The point (sin θ, cos θ), θ being any real number, lie inside the circle x 2 + y 2 – 2x – 2y + λ = 0, if
• The locus of the foot of perpendicular drawn from the centre of the ellipse x 2 + 3y 2 = 6 on any tangent to it is :
• The eccentricity of the conic defined by | (x-1)^ 2 +(y-2)^ 2 - (x-5)^ 2 +(y-5)^ 2 |=3 is :
• If x=9 is the chord of contact of the hyperbola x^ 2 -y^ 2 =9 , then the equation of the corresponding pair of tangent…
• The focus of the parabola is (1, 1) and the tangent at the vertex has the equation x + y = 1. Then which of the…
• The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the…
• A variable circle is drawn to pass through (1, 0) and to touch the line x+y=0 . Let S = 0 represent the locus of the…
• A line y=2 x+5 intersects a hyperbola only at a point (-2,1) . The equation of one of its asymptotes is 3 x+2 y+1=0 …
• Equation of a common tangents to the curves y 2 = 8x and xy = -1 is
• The points on the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 , such that the tangent at that point makes equal angles with…
• If one end of a focal chord AB of the parabola y^ 2 =8 x is at A ( 1 2 ,-2 ) , then the equation of the tangent to it…
• The intercept on the line y = x by the circle x^ 2 +y^ 2 -2 x=0 is AB. Equation of the circle on AB as a diameter is,
• The equation of a circle which passes through the point (1, –2) and (4, –3) and whose centre lies on the line 3x + 4 y…
• The name of the conic represented by the equation 13 x^ 2 -18 x y+37 y^ 2 +2 x+14 y-2=0 is,
• Equation of the ellipse whose foci are (4, 0) and (–4, 0) and e = 1/3 is
• The minimum distance of 4 x^ 2 +y^ 2 +4 x-4 y+5=0 from the line -4 x+3 y=3 is
• The intercept on the line y = x by the circle x 2 + y 2 – 2x = 0 is AB. Equation of the circle on AB as a diameter is
• The line 5 x+12 y=9 touches the hyperbola x^ 2 -9 y^ 2 =9 at the point,
• The locus of the centre of a circle, which touches externally the circle x 2 +y 2 –6x –6y +14 =0 and also touches the…
• A circle touches x axis at +3 distance and cuts an intercept of 8 in +ve direction of y axis. Its equation is–
• If p is the perpendicular from origin to the line x a + y b =1, then
• The equation x^ 2 10-a + y^ 2 4-a =1 represents an ellipse, if
• The distance of the line 2 x-3 y=4 from the point (1, 1) measured parallel to the line x+y=1 is,
• For the two circles x 2 + y 2 = 16 and x 2 + y 2 - 2y = 0, there is/are:
• The equation of the parabola having its axis parallel to x-axis and which passes through the points (1, 2), (–1,3) and…
• The equation of the line which is tangent to both the circle x^ 2 +y^ 2 =5 and the parabola y^ 2 =40 x is,
• If P=(1,0), Q=(-1,0) and R=(2,0) are three given points, then locus of the points satisfying the relation 5 Q^ 2 +5 R^…
• The equation of the line which passes through the point (3, 4) and the sum of its intercept on the axes is 14 is
• The equation 16x 2 – 3y 2 – 32x + 12y – 44 = 0 represents a hyperbola
• The common tangent to the circles x 2 + y 2 = 4 and x 2 + y 2 + 6x + 8y – 24 = 0 also passes through the point:
• The equation 9 x^ 2 -16 y^ 2 -18 x+32 y-151=0 represents,
• The slope of the common tangent to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 and a concentric circle of radius r is,
• If the algebraic sum of the perpendicular distance from the points (2, 0), (0, 2) and (1, 1) on a variable line is…
• The equation of the common tangent to the parabola y^ 2 =2 x and x^ 2 =16 y will be
• Let A(2, –3) and B(–2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on 2x + 3y = 1, then…
• If the circle x^ 2 +y^ 2 +2 x+2 k y+6=0 and x^ 2 +y^ 2 +2 k y+k=0 intersect orthogonally, then k is
• Equation of a common tangent to the parabola y^ 2 =4 x and the hyperbola xy = 2 is:
• The normal at the point (3, 4) on a circle cuts the circle at the point (-1, -2). Then the equation of the circle is,
• If a line is drawn through the origin and parallel to the line x – 2y + 5 = 0, then its eq. is
• The equation to the ellipse, whose focus is the point (–1, 1), whose directrix is the straight line x – y + 3 = 0, and…
• If the tangents on the ellipse 4 x^ 2 +y^ 2 =8 at the points (1, 2) and (a, b) are perpendicular to each other, then a…
• If P is a moving point in the xy–plane in such a way that perimeter of triangle PQR is 16 where Q ≡ (3, 5 ), R ≡ (7, 3…
• If hyperbola x^ 2 b^ 2 - y^ 2 a^ 2 =1 passes through the focus of ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 then eccentricity of…
• A 10 inches long pencil AB with mid point C and a small eraser P are placed on the horizontal top of a table such that…
• The equation of two straight lines through (7, 9) and making an angle of 60° with the line x- 3 y-2 3 =0 is :
• If the intercept made by the line between the axes is bisected at the point (x 1 , y 1 ), then its equation is
• The ratio in which the line segment joining the points (3, –4) and (–5, 6) is divided by the x-axis, is
• If two tangents drawn from a point P to the parabola ^ 2 = 4 are at right angles, then the locus of P is
• C is a point on the line segment joining A(–3, 4) and B (2, 1) such that AC = 2 BC then coordinates of C are
• P and Q are points on the line joining A(-2, 5) and B(3, 1) such that AP = PQ = QB then the mid point of PQ is -
• The locus of a point which moves such that the line segment having end points (2, 0) and (–2, 0) subtend a right angle…
• The equation of tangent drawn from the origin to the circle x 2 +y 2 –2rx –2hy + h 2 =0 is-
• Equation of the tangent to the hyperbola 2 x^ 2 -3 y^ 2 =6 which is parallel to the line y-3 x-4=0 is,
• The equation of the directrix of the parabola y 2 = 12 x is –
• If the line 3 x-2 y+p=0 is normal to the circle x^ 2 +y^ 2 =2 x-4 y-1 , then p will be
• The length of the latus-rectum of the parabola x = ay 2 + by + c is
• If θ and Φ are eccentric angles of the ends of a pair of conjugate diameters of the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 …
• The number of values of c such that the straight line y = 4x + c touches the curve x^ 2 4 +y^ 2 =1 is
• The range of values of θ ∈ [0, 2π] for which (1 + cos θ, sin θ) is on interior points of the circle x 2 + y 2 = 1, is
• If (1, 4) is the centroid of a triangle and two vertices are (4, –3) and (–9, 7), then third vertex is
• The angle between the tangents from (α, β) to the circle x 2 + y 2 = a 2 is –
• Equation of a common tangent with positive slope to the circle x^ 2 +y^ 2 -8 x=0 as well as to the hyperbola x^ 2 9 …
• Find the equation of the tangent to the hyperbola 4 x^ 2 -9 y^ 2 =1 which is parallel to the line 4 y=5 x+7 .
• If the foot of the perpendicular from the origin to a straight line is at the point (3, –4). Then, the equation of the…
• The sides AB, BC, CD and DA of a quadrilateral have the equations x + 2y = 3, x = 1, x – 3y = 4, 5x + y + 12 = 0…
• If the straight line through the point P(3, 4) makes an angle 6 with x-axis and meets the line 12x + 5y + 10 = 0 at Q…
• If P=(x, y), F_ 1 =(3,0), F_ 2 =(-3,0) and 16 x^ 2 +25 y^ 2 =400 , then P F_ 1 +P F_ 2 equals
• If the chord y = mx + 1 of the circle x 2 + y 2 = 1 subtends an angle of measure 45° at the major segment of the circle…
• Parametric equation of the ellipse x^ 2 16 + y^ 2 9 =1 is
• If the tangent at the point ( Zec Φ. Ztan Φ) of the hyperbola x^ 2 4 - y^ 2 9 =1 is parallel to 3 x-y+4=0 , then the…
• The equation of a tangent to the hyperbola 4 x^ 2 -5 y^ 2 =20 parallel to the line x-y=2 is:
• A point P moves on the line 2 x-3 y+4=0 . If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of…
• The ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 and the straight line y = mx + c intersect in real points only, if
• Let P = (–1, 0), Q = (0, 0) and R =(3,3 3 ) be three points. The equation of the bisector of the angle PQR is
• The line joining (5, 0) to (10 cosθ, 10 sinθ) is divided internally in the ratio 2 : 3 at P. If θ varies then the locus…
• Two common tangents to the circle x 2 + y 2 = 2a 2 and parabola y 2 = 8ax are
• If PQ is a focal chord of the parabola y^ 2 =4 a x, O is the origin and G is the centroid of triangle OPQ, then the…
• Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If…
• The equation of the tangent to the parabola ^ 2 = 4 a at the point (3, 2) is,
• The greatest distance of the point P (10,7) from the circle x^ 2 +y^ 2 -4 x-2 y-20=0 is
• Consider the ellipse x^ 2 3 + y^ 2 1 =1 . Let P, Q, R and S be four points on this ellipse such that the normals drawn…
• The equation of the normal to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 at the positive end of the latus rectum, is
• The point on the curve 3 x^ 2 -4 y^ 2 =72 which is nearest to the line 3 x+2 y+1=0 is,
• The value of m, for which the line y=m x+ 25 3 3 is a normal to the conic x^ 2 16 - y^ 2 9 =1 is,
• The parametric equation of the curve (y – 2) 2 = 12 (x – 4) are–
• A line has slope – 3/4, positive y-intercept and forms a triangle of area 24 sq. units with coordinate axes. Then, the…
• If tangents are drawn to the ellipse x^ 2 +2 y^ 2 =2 at all points on the ellipse other than its four vertices then the…
• The two circles x 2 + y 2 – 2x + 6y + 6 = 0 and x 2 + y 2 –5x + 6y + 15 = 0
• The value of x so that 2 is the slope of the line through (2,5) and (x, 3) is,
• The eccentricity of the ellipse which meets the straight line x 7 + y 2 =1 on the axis of x and the straight line x 3 …
• A point A(2, 1) is outside the circle x 2 + y 2 + 2gx + 2fy + c = 0 & AP, AQ are tangents to the circle. The equation…
• The eccentricity of the conjugate hyperbola of the hyperbola x 2 – 3y 2 = 1 is
• From a fixed point A, three normals are drawn to the parabola ^ 2 = 4 a at the points P, Q and R. Two circles C 1 and C…
• The points A(2a, 4a), B(2a, 6a) and C(2 a+ 3 a, 5 a)(a>0) are the vertices of
• Given below are two statements. One is labelled as Assertion and the other is labelled as Reason. STATEMENT - 1…
• PQ and RS are two perpendicular chords of the rectangular hyperbola x y=c^ 2 . If C is the centre of the rectangular…
• For the circle x 2 + y 2 +4x –7y +12 =0 the following statement is true-
• The two curves x^ 3 -3 x y^ 2 +2=0 and 3 x^ 2 y-y^ 3 -z=0
• The angle between the tangents to the circle x 2 + y 2 = 25 at (3, 4) and (4, –3) is :
• A triangle is formed by a tangent to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 and the coordinate axes. The area of triangle…
• A hyperbola passes through the point P( 2 , 3 ) and has foci at ( 2,0) . Then the tangent to this hyperbola at P also…
• If y = 2x is a chord of the circle x 2 + y 2 = 10x, then the equation of the circle whose diameter is this chord is
• The curve x y=c(c>0) and the circle x^ 2 +y^ 2 =1 touch at two points, then distance between the points of contact is
• P and Q are any two points on the circle x^ 2 +y^ 2 =4 such that PQ is a diameter. If and are the lengths of…
• The circles a x^ 2 +a y^ 2 +2 g_ 1 x+2 f_ 1 y+c_ 1 =0 and b x^ 2 +b y^ 2 +2 g_ 2 x+2 f_ 2 y+c_ 2 =0 (a 0 and b 0) cut…
• The centre of the circle passing through the point (0,1) and touching the curve y = x 2 at (2, 4) is :
• Any point on the parabola whose focus is (0, 1) and the directrix is x + 2 = 0 is given by
• The straight lines x+y=0,3 x+y-4=0 and x+3 y-4=0 form a triangle which is,
• If two distinct chords, drawn from the point ( p, q ) on the circle x 2 + y 2 = px + qy (where pq ≠ 0) are bisected by…
• An ellipse, with foci, at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?
• The equation of the hyperbola in the standard form (with transverse axis along the x-axis) having the length of the…
• The slope of the line touching both the parabolas y 2 = 4x and x 2 = –32y is :
• The locus of the point of intersection of the lines and 3 k+k-4 3 =0 for different values of k is-
• The equation of a line parallel to ax + by + c = 0 and passing through the point (c, d) is
• The equation of the straight line which passes through the intersection of the lines x – y – 1 = 0 and 2x – 3y + 1 = 0…
• Centre of the circle x 2 + y 2 – 2x + 4y + 1 = 0 is –
• The tangent to the parabola y^ 2 =4 x at the point where it intersects the circle x^ 2 +y^ 2 =5 in the first quadrant…
• If y= x-3, y= x+1, y=x+4 are three normals drawn from a fixed point P to the parabola whose axis along x-axis, then 2…
• The equation of the straight line which passes through the point of intersection of the straight lines x + 2y = 5 and…
• Equation of the circle of radius 5 whose centre lies on y-axis in first quadrant and passes through (3, 2) is
• If the expression a x^ 2 +b y^ 2 +c z^ 2 +2 a y z+2 b z x+2 c x y can be resolved into rational factors, then
• Find the area of region bounded by the curve x^ 2 -4 x+y^ 2 =0 .
• The equation to a circle passing through the origin and cutting of intercepts each equal to 5 of the axes is-
• The equation of the common tangents to the parabola y^ 2 =8 x and hyperbola 3 x^ 2 -y^ 2 =3 is,
• Let the line y = mx and the ellipse 2 x^ 2 +y^ 2 =1 intersect at a point P in the first quadrant. If the normal to this…
• If 5 x+9=0 is the directrix of the hyperbola 16 x^ 2 -9 y^ 2 =144 , then its corresponding focus is
• The lines 8x + 4y = 1, 8x + 4y = 5, 4x + 8y = 3, 4x + 8y = 7 form a
• The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle x 2 + y 2 = 9 is :
• The incentre of the triangle formed by the lines y = 15, 12y = 5x and 3x + 4y = 0 is
• If the two lines x+a-1) y=1 and 2 x+a^ 2 y=1(a R-(0,1) are perpendicular, then the distance of their point of…
• One of the focus of the hyperbola 3(y – 1) 2 – 4 ( x – 2) 2 = 12 is
• The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, -3), then its…
• The equation ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 0 represents a circle only if
• Equation of parabola which has its axis along x-axis and which passes through the points (3, 2) and (–2, –1) is
• The curve represented by x = 2 (cos t + sin t), y = 5 (cos t – sin t) is
• The ellipse x^ 2 25 + y^ 2 16 =1 and the hyperbola x^ 2 25 - y^ 2 16 =1 have in common
• The name of the curve described parametrically by the equations x = t ^ 2 + t +1_ 1 y = t ^ 2 - t +1 , is
• If the chords of contact of tangents from two points [x_ 1 , Y_ 1 ] and [x_ 2 , y_ 2 ] to the ellipse x^ 2 a^ 2 + y^ 2…
• If the vertex = (2, 0) and the extremities of the latus rectum are (3, 2) and (3, –2), then the equation of the…
• If the vertices P, Q, R of a Δ PQR are rational points, which of the following points of the Δ PQR is/are always…
• The equation of a line through the point (–4, –3) and parallel to x-axis, is
• A non-horizontal line passing through the point (4, –2) whose distance from origin is 2 units is
• If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of…
• The equation of the ellipse which passes through origin and has its foci at the points (1, 0) and (3, 0) is -
• The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola x^ 2 4 - y^ 2 5 =1, meets x-axis…
• A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the…
• Two points (a, 0) and (0, b) are joined by a straight line. Another point on this line is
• The number of values of c such that the straight line y=4 x+c touches the curve x^ 2 4 +y^ 2 =1 is,
• The eccentricity of an ellipse whose centre is at the origin is 1 2 . If one of the directrices is x = 4, then the…
• A line passes through the point of intersection of the lines 100x + 50y – 1 = 0 and 75x + 25y + 3 = 0 and makes equal…
• The coordinates of a point on x + y + 3 = 0, whose distance from x + 2y + 2 = 0 is 5 is equal to -
• The equation of the conjugate hyperbola of the hyperbola 3 x^ 2 -5 x y-2 y^ 2 +5 x+11 y-8=0 is,
• If the latus rectum of an ellipse is half of its minor axis, its eccentricity is
• The equation x^ 2 1-r - y^ 2 1+r =1,|r|<1 represents
• Equation of ellipse whose minor axis is equal to the distance between the foci and whose latus rectum is 10, is given…

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