Co-ordinate Geometry — practice questions

254 curated practice questions from Co-ordinate Geometry (Mathematics JEE Advanced, NTA Exams) — 206 hard, 48 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• An ellipse with major axis 4 and minor axis 2 touches both the coordinate axis, then locus of its centre is
• If the equation, 2x 2 + k xy – 3y 2 – x – 4y – 1 = 0 represents a pair of lines then the value of k can be :
• If P is a variable point and F 1 and F 2 are two fixed points such that |PF 1 – PF 2 | = 2a. Then the locus of the…
• The equation x^ 2 1-r - y^ 2 1+r = 1, |r| < 1 represents
• If θ is the angle between the lines represented by x 2 – 3xy + λy 2 + 3x – 5y + 2 = 0, where λ is a real number, then…
• The diagonals of a parallelogram PQRS are along the lines x + 3 y = 4 and 6 x – 2 y = 7. Then PQRS must be a
• The equation of tangents to the ellipse 9x 2 + 16y 2 = 144 which pass through the point (2, 3)
• The circle described on the line joining the points (0, 1), (a, b) as diameter cuts the x-axis in points whose…
• A right angle triangle ABC having ‘C’ a right angle has AC = a and BC = b units. The points A and B slide along the…
• The equation of the circle whose radius is 3 and which touches the circle x 2 + y 2 – 4x – 6y – 12 = 0 internally at…
• Let AB be a chord of the circle x 2 + y 2 = r 2 subtending a right angle at the centre. Then the locus of the centroid…
• The coordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2) then the coordinates of…
• If the radius of the circumcircle of the triangle TPQ, where PQ is chord of contact corresponding to point T with…
• A line meets the co-ordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d 1 and d 2 are the…
• The points , (1, 3) and (82, 30) are vertices of
• If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is
• Length of the focal chord of the parabola y 2 = 4ax at a distance p from the vertex is
• T is a point on the tangent to a parabola y 2 = 4ax at its point P. TL and TN are the perpendiculars on the focal…
• If A (3,4) and B is a variable point on the lines |x| = 6. If AB 4 then the number of positions of B with integral…
• The correct matching is :
• A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at A , B and C . If the centroid D (…
• Given the family of lines, a (3x +4y +6) + b (x +y +2) =0 . The line of the family situated at the greatest distance…
• On the line segment joining (1, 0) and (3, 0) an equilateral triangle is drawn having its vertex in the fourth…
• Assertion : Two orthogonal circles intersect to generate a common chord which subtends complimentary angles at their…
• The equation of the circle passing through (1, 1) and the points of intersection of x 2 + y 2 + 13 x – 3y = 0 and 2x 2…
• If P, Q, R are three co-normal points on the parabola y 2 = 4ax then the centroid of the triangle PQR always lies on :
• The length of the common chord of the circles (x – a) 2 + y 2 = c 2 and x 2 + (y – b) 2 = c 2 is-
• If α 1 , α 2 , α 3 , β 1 , β 2 , β 3 are the values of n for which _ r=0 ^ n-1 x^ 2 r is divisible by then the triangle…
• If the circle x 2 + y 2 + 2x + 2ky + 6 = 0 and x 2 + y 2 + 2ky + k = 0 intersect orthogonally, then k is :
• Let OX and OY be two fixed lines inclined at a constant angle α. A variable line cuts OX at P and OY at Q. From P and Q…
• The circle x² + y² – 6x – 10y + c = 0 does not intersect or touch either axis & the point (1, 4) is inside the circle…
• The eccentricity of the conic 4(2y – x – 3) 2 – 9 (2x + y – 1) 2 = 80 is
• If a circle passes through the point (1, 2) and cuts the circle x 2 + y 2 = 4 orthogonally, then the locus of its…
• If α + β = 3π then the chord joining the points α and β for the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 =1 passes through
• If P (1, 2), Q (4,6), R (5, 7) and S ( a , b ) are the vertices of a parallelogram PQRS , then
• The difference of the slopes of the lines, x 2 (sec 2 θ – sin 2 θ) – 2xy tan θ + y 2 sin 2 θ = 0 is
• The straight lines x + y = 0, 3x + y – 4 = 0, x + 3y – 4 = 0 form a triangle which is :
• A ray of light travels along a line y = 4 and strikes the surface of a curve y 2 = 4(x + y) then equation of the line…
• Let denote the set of all real numbers. Then the area of the region array r (x, y) R imes R : x>0, y> 1 x , 5 x-4…
• If the circle x 2 + y 2 – 2x – 2y + 1 = 0 is inscribed in a triangle whose two sides are axes and one side has negative…
• Coordinates of the centre of a circle, whose radius is 2 unit and which touches the line pair x 2 – y 2 – 2x + 1 = 0…
• The length of the tangent drawn from any point on the circle x² + y² + 2gx + 2fy + p = 0 to the circle x 2 + y 2 + 2gx…
• Drawn from the origin are two mutually perpendicular straight lines forming an isosceles triangle together with the…
• A variable straight line passes through a fixed point (a, b) intersecting the co-ordinates axes at A and B. If ‘O’ is…
• Assertion : The equations of the straight lines joining origin to the points of intersection of x 2 + y 2 – 4x – 2y = 4…
• If equation of tangent at P, Q and vertex A of a parabola are 3x + 4y – 7 = 0, 2x + 3y – 10 = 0 and x – y = 0…
• Consider the straight line ax + by = c, where a, b, c ∈ R + this line meets the coordinate axes at A and B…
• Assertion : In a triangle ABC, if base BC is fixed and perimeter of the triangle is constant, then vertex A moves on an…
• The equation of the line segment AB is y = x. If A and B lie on the same side of the line mirror 2x – y = 1, the image…
• The area of the parallelogram formed by the lines 3x - 4y + 1 = 0, 3x - 4y + 3 = 0, 4x - 3y - 1 = 0 and 4x - 3y - 2 =…
• Assertion : Each point on the line y – x + 12 = 0 is equidistant from the lines 4y + 3x – 12 = 0, 3y + 4x – 24 =0 …
• Assertion : If straight line x = 8 meets the parabola y 2 = 8x at P & Q then PQ substends a right angle at the origin…
• If the vertices P , Q , R of a triangle PQR are rational points, which of the following points of the triangle PQR is…
• If the line x + my + am 2 = 0 touches the parabola y 2 = 4ax, then the point of contact is
• The equations to a pair of opposite sides of parallelogram are x 2 – 5 x + 6 = 0 and y 2 – 6 y + 5 = 0, the equations…
• Normals at three points P, Q, R at the parabola y 2 = 4ax meet in a point A and S be its focus, if |SP|. |SQ| . |SR| =…
• If tangent at (1, 2) to the circle c 1 : x 2 + y 2 = 5 intersects the circle c 2 : x 2 + y 2 = 9 at A and B and…
• If the tangents are drawn from any point on the line x + y = 3 to the circle x 2 + y 2 = 9, then the chord of contact…
• Let 0 P = (cos θ, sin θ) and Q = (cos (α – θ), sin (α – θ)) then θ is obtained from P by
• Assertion : The perpendicular bisector of the line segment joining the point (–a, 2 at) and (a, 0) is tangent to the…
• The area of the region ( x, y ) : 0 ≤ x ≤ 0 ≤ y ≤ 1, x ≥ 3 y , x + y ≥ 2 is
• A circle touches a straight line l x + my + n = 0 & cuts the circle x 2 + y 2 = 9 orthogonally. The locus of centres of…
• The number of values of k , for which the system of equations ( k + 1) x + 8 y = 4 k, kx + ( k + 3) y = 3 k – 1 has…
• Assertion : Feet of perpendiculars drawn from foci of an ellipse 4x 2 + y 2 = 16 on the line 2 3 x + y = 8 lie on the…
• From a point R(5, 8) two tangents RP and RQ are drawn to a given circle S = 0 whose radius is 5. If circumcentre of the…
• If the normals at two points P, Q of the parabola, y 2 = 4x intersect at a third point R on the parabola, then the…
• From an external point P, pair of tangent lines are drawn to the parabola, y 2 = 4x. If θ 1 and θ 2 are the…
• If the normals drawn from any point to the parabola y 2 = 4ax cut the line x = 2a in points whose ordinates are in…
• From the point (15, 12) three normals are drawn to the parabola y 2 = 4x, then centroid of triangle formed by three…
• The locus of the mid point of the chords of the circle x 2 + y 2 = a 2 , which are tangent to the hyperbola x^ 2 a^ 2 …
• If A & B are points on the parabola y 2 = 4ax with vertex O such that OA perpendicular to OB & having lengths r 1 & r 2…
• The area of a triangle formed by the lines x – y = 0, x + y = 0 and any tangent to the hyperbola x 2 – y 2 = a 2 is
• The equation of the chords of length 5 and passing through the point (3, 4) on the circle 4x 2 + 4y 2 – 24x – 7y = 0 are
• Two circles passing through A(1, 2) and B(2, 1) touch the lines 4x + 8y – 7 = 0 at C and D such that ACED is…
• If the normal to the parabola y 2 = 4ax at the point (at 2 , 2at) cuts the parabola again at (aT 2 , 2aT), then
• Let S denote the locus of the mid-points of those chords of the parabola y 2 = x , such that the area of the region…
• Consider a circle with its centre lying on the focus of the parabola, y 2 = 2 px (p > 0) such that it touches the…
• All points lying inside the triangle formed by the points (1, 3), (5, 0) and (–1, 2) satisfy
• Assertion : Number of common tangents of x 2 + y 2 – 2x – 4y – 95 =0 and x 2 + y 2 – 6x – 8y + 16 =0 is zero. Reason …
• The points (–a, –b), (0, 0), (a, b) and (a 2 , ab) are
• Let u x + y + a [3] b =0 v k - d + b [3] a =0 a, b ∈ R be two straight lines. The equation of the bisectors of the…
• Through the vertex O of the parabola y 2 = 4ax two chords OP & OQ are drawn and the circles on OP & OQ as diameter…
• Let x and y be the real numbers satisfying the equation x² – 4x +y² +3 =0. If the maximum and minimum values of x² + y²…
• Let A (0, β ), B (–2, 0) and C (1, 1) be the vertices of a triangle then If I 1 is the interval of values of β for…
• The circumference of the circle x 2 + y 2 – 2x + 8y – q = 0 is bisected by the circle x 2 + y 2 + 4x + 12y + p = 0…
• Column II contains value of k according to condition given in Column I Which of the following option is incorrect
• If the straight line x + y = 1 is a normal to the parabola x 2 = ay, then the value of a is
• If P (1, 2), Q (4, 6), R (5, 7) and S (a, b) are the vertices of a parallelogram PQRS,then:
• Line L has intercepts a and b on the coordinate axes. When the axes are rotated through given angle, keeping the origin…
• Let P ( a secθ, b tanθ) and Q ( a secφ, b tanφ), where θ + φ = , be two points on the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2 =…
• The equation, 3x 2 + 4y 2 – 18x + 16y + 43 = c.
• The number of common tangents to the circles x 2 +y 2 = 4 and x 2 + y 2 – 6x – 8y = 24 is :
• Equation of a straight line passing through the point (4, 5) and equally inclined to the lines, 3x = 4y + 7 and 5y =…
• AB is a diameter of a circle and C is any point on the circumference of the circle. Then :
• The length of the side of an equilateral triangle inscribed in the parabola, y 2 = 4x so that one of its angular point…
• If (5, 12) and (24, 7) are the foci of a conic passing through the origin then the eccentricity of conic is
• Let PS be the median of the triangle with vertices P (2, 2), Q (6, –1) and R (7, 3). The equation of the line passing…
• The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of…
• The equation of the circle which is touched by y = x, has its centre on the positive x-axis & cuts off a chord of…
• Let A (0, β), B (–2, 0) and C (1, 1) be the vertices of a triangle then Angle A of the triangle ABC will be obtuse if β…
• Tangent is drawn to ellipse x^ 2 27 +y^ 2 =1 at (3 3 , ) (where θ ∈ (0, π/2)). Then the value of θ such that sum of…
• A straight line L through the point (3, –2) is inclined at an angle 60° to the line If L also intersects the x -axis…
• A circle passes through the point (3, 7 2 ) and touches the line pair x 2 – y 2 – 2x + 1 = 0 . The co-ordinates of the…
• P is a variable point on the line L = 0 . Tangents are drawn to the circle x 2 + y 2 = 4 from P to touch it at Q and R…
• Two thin rods AB & CD of lengths 2a & 2b move along OX & OY respectively, when ‘O’ is the origin. The equation of the…
• If r 1 and r 2 be the radius of smallest and largest circle which passes through (5, 6) and touches the circle (x – 2)…
• If a chord of the circle x 2 + y 2 = 8 makes equal intercepts of length a on the coordinate axes, then-
• Let x, y be real variable satisfying the x 2 + y 2 + 8x – 10y – 40 = 0 . Let a = max (x+2^ 2 )+(y-3)^ 2 and b = min …
• Let P = (1, 1) and Q = (3, 2). The point R on the x-axis such that PR + RQ is the minimum is
• If the straight line ax + by = 2, b ≠ 0, touches the circle x 2 + y 2 – 2x = 3 and is normal to the circle x 2 + y 2 –…
• If the squares of the lengths of the tangents from a point P to the circles x 2 + y 2 = a 2 , x 2 + y 2 = b 2 and x 2 +…
• If two normals to a parabola y 2 = 4ax intersect at right angles then the chord joining their feet passes through a…
• The equations of the sides of a square whose each side is of length 4 units and centre is (1, 1). Given that one pair…
• A circle is described whose centre is the vertex and whose diameter is three–quarters of the latus rectum of the…
• If pairs of straight lines, x 2 – 2pxy – y 2 = 0 and x 2 – 2q xy – y 2 = 0 be such that each pair bisects the angles…
• 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
• From the focus of the parabola, y 2 = 8x as centre, a circle is described so that a common chord of the curves is…
• The abscissa of two points A and B are the roots of the equation x 2 + 2ax – b 2 = 0 and their ordinates are the roots…
• The orthocentre of the triangle ABC is ‘B’ and the circumcentre is ‘S’ (a, b). If A is the origin then the co–ordinates…
• If P is a variable point and F 1 and F 2 are two fixed points such that |PF 1 – PF 2 | = 2a. Then the locus of the…
• ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies…
• The correct matching is :
• The straight lines x + y = 0, 3 x + y – 4 = 0, x + 3 y – 4 = 0 form a triangle which is
• A system of circles is said to be coaxial when every pair of the circles has the same radical axis. It follows from…
• If tan θ 1 tan θ 2 = – a ^ 2 ~b ^ 2 , then the chord joining two point θ 1 and θ 2 on the ellipse x^ 2 a^ 2 + y^ 2 b^ 2…
• Let PQR be right angled isosceles triangle, right angled at P (2,1). If the equation of the line QR is 2 x + y = 3…
• If L ≡ 2x + y – 6 = 0, then the locus of circumcentre of ∆PQR is P is a variable point on the line L = 0 . Tangents are…
• If foci of x^ 2 a^ 2 - y^ 2 b^ 2 =1 coincide with the foci of x^ 2 25 + y^ 2 9 =1 and eccentricity of the hyperbola is…
• If the distance of 2 points P and Q on parabola y 2 = 4ax from the focus are 4 and 9 respectively, then the distance of…
• The normal chord at a point ‘t’ on the parabola y 2 = 4ax subtends a right angle at the vertex. Then t² is equal to :
• If C 1 : x 2 + y 2 = (3 + 2 2 ) 2 be a circle and PA and PB are pair of tangents on C 1 where P is any point on the…
• If P = (1, 0), Q = (–1, 0) and R = (2, 0) are three given points, then locus of the point S satisfying the relation SQ…
• The equation of the common tangents to the parabola y 2 = 8x and the hyperbola 3x 2 – y 2 = 3 is-
• The area bounded by the curves y = | x |–1 and y = –| x |+1 is
• The correct matching is :
• If P is a variable point and F 1 and F 2 are two fixed points such that |PF 1 – PF 2 | = 2a. Then the locus of the…
• Tangents are drawn to y^ 2 =4 a x from a variable point P moving on x+a=0, then the locus of foot of perpendicular…
• A variable circle having fixed radius ‘a’ passes through origin and meets the coordinate axes in point A and B. Locus…
• Area of the parallelogram formed by the lines y = mx , y = mx + 1, y = nx and y = nx + 1 equals
• The point (4, 1) undergoes the following three transformations successively. (I) Reflection about the line y = x (II)…
• Let S = ( x , y )∈ × : x ≥ 0, y ≥ 0, y 2 ≤ 4 x , y 2 ≤ 12 – 2 x and 3 y+ 8 x 5 8 . If the area of the region S is 2 …
• Let P = (–1, 0), Q = (0, 0) and be three points. Then the equation of the bisector of the angle PQR is
• The set of values of ‘b’ for which the origin and the point (1, 1) lie on the same side of the straight line a 2 x +…
• If the locus of the circumcentre of a variable triangle having sides y–axis, y = 2 and l x + my = 1, where ( l, m) lies…
• The point (4, 1) undergoes the following three transfor-mations successively (i) reflection about the line y = x (ii)…
• A system of circles is said to be coaxial when every pair of the circles has the same radical axis. It follows from…
• The equation 9x 2 – 16y 2 – 18x + 32y – 151 = 0 represents a hyperbola -
• Area of the quadrilateral formed by the lines | x | + | y | = 2 is :
• The area bounded by the angle bisectors of the lines x 2 – y 2 + 2 y = 1 and the line x + y = 3, is
• The locus of the middle points of the focal chords of the parabola, y 2 = 4x is :
• A straight line through the vertex P of a triangle PQR, intersects the side QR at the point S and the circum-circle of…
• The equation of the bisector of the angle between two lines 3x – 4y + 12 = 0 and 12x – 5y + 7 = 0 which contains the…
• Let A (0, β), B (–2, 0) and C (1, 1) be the vertices of a triangle then All the values of β for which angle A of the…
• Consider three planes P 1 : x – y + z = 1 P 2 : x + y – z = –1 P 3 : x – 3 y + 3 z = 2 . Let L 1 , L 2 , L 3 be the…
• The locus of the orthocentre of the triangle formed by the lines (1 + p ) x – py + p (1 + p ) = 0, (1 + q ) x – qy + q…
• Let OX and OY be two fixed lines inclined at a constant angle α. A variable line cuts OX at P and OY at Q. From P and Q…
• Orthocentre of triangle with vertices (0, 0), (3, 4) and (4, 0) is
• The correct matching is :
• For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay +…
• A light beam emanating from the point A(3, 10) reflects from the straight line 2x + y – 6 = 0 and then passes through…
• P is a variable point on the line L = 0 . Tangents are drawn to the circle x 2 + y 2 = 4 from P to touch it at Q and R…
• The point (a 2 , a + 1) is a point in the angle between the lines 3x – y + 1 = 0 and x + 2y – 5 = 0 containing the…
• The distance of a point on the ellipse x^ 2 6 + y^ 2 2 =1 from the centre is 2. The eccentric angle of the point is
• Let OX and OY be two fixed lines inclined at a constant angle α. A variable line cuts OX at P and OY at Q. From P and Q…
• The distance between the lines represented by the equation, x^ 2 +2 2 y+2 y^ 2 +4 x+4 2 y+1=0 is :
• Assertion : Let S 1 : x 2 + y 2 – 10x – 12y – 39 = 0 S 2 : x 2 + y 2 – 2x – 4y + 1 = 0 S 3 : 2x 2 + 2y 2 – 20x – 24y +…
• If the chord through the points whose eccentric angles are θ & φ on the ellipse, x^ 2 a^ 2 + y^ 2 b^ 2 =1 passes…
• AB is a diameter of a circle and C is any point on the circumference of the circle. Then
• The angle of intersection of the two circles x 2 + y 2 – 2x – 2y = 0 and x 2 + y 2 = 4, is–
• Two tangents to the parabola y 2 = 4ax make angle α 1 and α 2 with the x-axis. The locus of their point of intersection…
• The triangle PQR of area ‘A’ is inscribed in the parabola y 2 = 4ax such that the P lies at the vertex of the parabola…
• Match the items of Column I with those of Column II. The correct matching is :
• The co-ordinates of a point P on the line 2x – y + 5 = 0 such that |PA – PB| is maximum where A is (4, –2) and B is (2…
• Consider the lines L_ 1 : x+1 3 = y+2 1 = z+1 2 The distance of the point (1, 1, 1) from the plane passing through the…
• Two vertices of a triangle are (4, –3) and (–2, 5). If the othocentre of the triangle is at (1, 2), then the…
• If the locus of the circumcentre of a variable triangle having sides y–axis, y = 2 and l x + my = 1, where ( l, m) lies…
• The locus of the mid point of the focal radii of a variable point moving on the parabola, y 2 = 4ax is a parabola whose
• The equation of a circle which touches x-axis and the line 4x –3y +4 =0, its centre lying in the third quadrant and…
• If the area of the quadrilateral formed by the tangents from the origin to the circle x 2 + y 2 + 6x – 10y + C = 0 and…
• If P is a point of the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1, whose foci are S and S’. Let ∠PSS’ = α and ∠PS’S= β, then
• On the portion of the straight line, x + 2y = 4 intercepted between the axes, a square is constructed on the side of…
• AB, AC are tangents to a parabola y 2 = 4ax. p 1 , p 2 and p 3 are the lengths of the perpendiculars from A, B and C…
• The condition that the line, x. cosθ + y. sin θ =p touches the parabola, y 2 = 4a (x + a) is :
• Tangent are drawn from the points on the line x – y – 5 = 0 to x 2 + 4y 2 = 4, then all the chords of contact pass…
• All the points lying on or inside the triangle formed by the points (1, 3), (5, 6) and (–1, 2) satisfy
• If sum and product of the slopes of lines represented by 4x 2 + 2hxy – 7y 2 = 0 is equal then h is equal to :
• A line cuts x–axis at A and y–axis at B such that AB = l . Column II contain equation of locus of point given in Column…
• Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r . If PS and RQ intersect at a…
• Equation of chord of the circle x 2 + y 2 – 3x – 4y – 4 = 0 which passes through the origin such that origin divides it…
• The orthocentre of the triangle formed by the lines x = 0, y = 0 and x + y = 1 is
• From the point A(0, 3) on the circle x² + 4x + (y – 3)² = 0, a chord AB is drawn & extended to a point M such that AM =…
• The equations of three lines, AB, CD and EF are, (b – c) x + (c – a) y + (a – b) = 0, (c – a) x + (a – b) y + (b – c) =…
• Find the angle between the lines represented by the equation x 2 – 2pxy + y 2 = 0
• If x c + y d =1 is a line through the intersection of x a + y b =1 and x ~b + y a =1 and the lengths of the…
• The points (– a , – b ), (0, 0), ( a , b ) and ( a 2 , ab ) are
• If M is the foot of the perpendicular from a point P of a parabola y 2 = 4ax to its directrix and SPM is an equilateral…
• Assertion : If (a 1 x + b 1 y + c 1 ) + (a 2 x + b 2 y + c 2 ) + (a 3 x + b 3 y + c 3 ) = 0, then lines a 1 x + b 1 y +…
• Let 3 denote the three-dimensional space. Take two points P = (1, 2, 3) and Q = (4, 2, 7). Let dist ( X , Y ) denote…
• Let V be the vertex and L be the latus rectum of the parabola x 2 = 2y + 4x – 4 . Then the equation of the parabola…
• Let P 1 and P 2 be two planes given by P 1 : 10 x + 15 y + 12 z – 60 = 0, P 2 : –2 x + 5 y + 4 z – 20 = 0 . Which of…
• The points (s) on the parabola y 2 = 4x which are closest to the circle : x 2 + y 2 – 24y + 128 = 0 is/are
• Let L 1 and L 2 be the following straight lines. L_ 1 : x-1 1 = y -1 = z-1 3 and L_ 2 : x-1 -3 = y -1 = z-1 1 . Suppose…
• Let P(a secθ, b tanθ) and Q (a sec φ, b tan φ), where θ + φ = 2 , be two points on the hyperbola x^ 2 a^ 2 - y^ 2 b^ 2…
• The equation of common tangent to the parabola, y 2 = 2x and x 2 = 16y is : Ax + By + C = 0, where A, B, C ∈ N, such…
• The equation of the other normal to the parabola y 2 = 4ax which passes through the intersection of those at (4a, –4a)…
• If the sum of the distance of a point from two perpendicular lines in a plane is 1, then its locus is :
• The ends of a line segment are P (1, 3) and Q (1, 1). R is a point on the line segment PQ such that PR : QR = 1 : λ. If…
• The vertices of ∆OBC are respectively (0, 0), (–3, –1) and (–1, –3). The equation of line parallel to BC and at a…
• Let O be the origin and let PQR be an arbitrary triangle. The point S is such that Then the triangle PQR has S as its
• Assertion : The diagonals of parallelogram formed by the lines ax + by + c = 0, a´x + b´y + c = 0, ax + by + c´ = 0…
• If the locus of the circumcentre of a variable triangle having sides y–axis, y = 2 and l x + my = 1, where ( l, m) lies…
• The two parabola y 2 = 4ax and y 2 = 4c (x –b) cannot have a common normal, other than the axis unless, if
• If the circle C 1 : x 2 + y 2 = 16 intersects another circle C 2 of radius 5 in such a manner that the common chord is…
• If a circle passes through the point (a, b) & cuts the circle x 2 + y 2 = K 2 orthogonally, then the equation of the…
• The line x + 3y – 2 = 0 bisects the angle between a pair of straight lines of which one has equation x – 7y + 5 =0 …
• A is a point on either of two lines y+ 3 |x|=2 at a distance of units from their point of intersection. The…
• Find the condition that the pair of straight lines joining the origin to the intersections of the line y = mx + c and…
• The locus of the foot of the perpendiculars drawn from the vertex on a variable tangent to the parabola y 2 = 4ax is :
• Let g ( x ) be a function defined on [–1, 1]. If the area of the equilateral triangle with two of its vertices at (0…
• The base BC of a ∆ABC is bisected at the point (p, q) and the equation to the side AB and AC are px + qy = 1 and qx +…
• The value of ‘a’ for which the common chord of the circles C 1 : x 2 + y 2 = 8 and C 2 : (x – a) 2 + y 2 = 8 subtends a…
• The triangle PQR is inscribed in the circle x 2 + y 2 = 25. If Q and R have coordinates (3, 4) and (–4, 3)…
• A hyperbola has centre 'C' and one focus at P(6,8) . If its two directrices are 3 x+4 y+10=0 and 3 x+4 y-10=0 then C P=
• Assertion : The four straight lines given by 12x 2 + 7xy – 12y 2 = 0 and 12x 2 +7xy –12y 2 –x +7y –1 =0 are the sides…
• Point M moved on the circle (x – 4) 2 + (y – 8) 2 = 20 . Then it broke away from it and moving along a tangent to the…
• Consider the circle x 2 + y 2 – 10x – 6y + 30 = 0 . Let O be the centre of the circle and tangent at A(7, 3) and B(5…
• Given the points A (0, 4) and B (0,–4), the equation of the locus of the point P (x, y) such that |AP – BP| = 6 is :
• The triangle formed by the tangent to the curve f ( x ) = x 2 + bx – b at the point (1, 1) and the coordinate axes…
• Assertion : Circumcircle of a triangle formed by the lines x = 0, x + y + 1 = 0 & x – y + 1 = 0 also passes through the…
• A normal is drawn to the parabola y 2 = 4ax at the point (2a, –2 2 a) then the length of the normal chord, is
• The graph of the function cos x cos (x + 2) – cos 2 (x + 1) is
• A straight line through the origin meets the parallel lines 4 x + 2 y = 9 and 2 x + y + 6 = 0 of a points P and Q…
• A tangent to the parabola x 2 + 4ay = 0 cuts the parabola x 2 = 4by at A and B the locus of the mid point of AB is :
• The bisector of the acute angle between the lines 3x – 4y + 7 = 0 and 12x + 5y – 2 = 0, is :
• Minimum radius of circle which is orthogonal with both the circle x 2 + y 2 –12x + 35 =0 and x 2 + y 2 + 4x +3 =0 is
• The acute angle between two straight lines passing through the point M(–6, –8) and the points in which the line segment…
• Minimum area of the triangle formed by any tangent to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1 with the coordinate axes is
• The centre of the smallest circle touching the circles x 2 + y 2 – 2y – 3 = 0 and x 2 + y 2 – 8x – 18y + 93 = 0 is :
• The three lines px + qy + r = 0, qx + ry + p = 0, rx + py + q = 0 are concurrent if
• Two circles x 2 + y 2 = 6 and x 2 + y 2 – 6x + 8 = 0 are given. Then the equation of the circle through their points of…
• 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…
• If the two circles, x 2 + y 2 + 2g 1 x + 2f 1 y = 0 and x 2 + y 2 + 2g 2 x + 2f 2 y = 0 touch each other then
• A rhombus is inscribed in the region common to the two circles x 2 + y 2 – 4x – 12 = 0 and x 2 + y 2 + 4x – 12 = 0 with…
• A triangle ABC with vertices A(–1, 0), B (–2, 3/4) and C (–3, –7/6) has its orthocentre H. Then the orthocentre of…
• Which of the following lines, is a normal to the parabola y 2 = 16 x
• Assertion : for two non–intersecting circles, direct common tangents subtends a right angle at either of point of…
• Given two straight lines x – y – 7 = 0 and x – y + 3 = 0. Equation of a line which divides the distance between them in…
• Let PQR be a right angled isosceles triangle, right angled at P (2, 1). If the equation of the line QR is 2x + y = 3…
• A line is drawn through the point A (p, q) in the direction θ to meet the line, ax + by + c = 0 in B, then | AB | =
• Given the four lines with the equations x + 2y – 3 = 0, 3x + 4y – 7 = 0, 2x + 3y – 4 = 0, 4x + 5y – 6 = 0 then :
• The normal to the curve x = a (cos θ + θ sin θ) y = a (sin θ – θ cos θ) at any point θ is such that
• The value of ‘c’ for which the set, (x, y) | x²+y² +2x 0 contains only one point in common is :
• A system of circles is said to be coaxial when every pair of the circles has the same radical axis. It follows from…
• The distance from the centre of the circle x² + y² = 2x to the straight line passing through the points of intersection…

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