If P is a variable point and F 1 and F 2 are two fixed points such that |PF 1 – PF 2 | =…

If P is a variable point and F₁ and F₂ are two fixed points such that |PF₁ – PF₂| = 2a. Then the locus of the point P is a hyperbola, with points F₁ and F₂ as the two focii (F₁F₂ > 2a). If \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is a hyperbola, then its conjugate hyperbola is \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1 .\) Let P(x, y) is a variable point such that \(\left|\sqrt{(x-1)^{2}+(y-2)^{2}}-\sqrt{(x-5)^{2}+(y-5)^{2}}\right|=3 .\)
If origin is shifted to point \(\left(3, \frac{7}{2}\right)\) and the axes are rotated through an angle θ in clockwise sense so that equation of given of given hyperbola changes to the standard form \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1,\) then θ is :