question_answer

Let AB be the chord of contact of the point \[(3,-3)\] w.r.t. the circle\[{{x}^{2}}+{{y}^{2}}=9\]. Then the locus of the orthocentre of \[\Delta PAB,\]where P be any point moving on the circle, is                                                                                  

A) \[{{(x-3)}^{2}}+{{(y+3)}^{2}}=9\]   

B) \[{{(x-3)}^{2}}+{{(y+3)}^{2}}=9/2\]

C) \[{{(x+3)}^{2}}+{{(y-3)}^{2}}=9\]

D) \[{{(x+3)}^{2}}+{{(y-3)}^{2}}=9/2\]

Correct Answer: A

Solution :

[a] The equation of AB is \[x-y=3\](chord of contact). Let orthocentre be \[H(h,k)\]. Its image w.r.t. AB is \[Q(k+3,h-3)\] Q will lie on the circle \[{{x}^{2}}+{{y}^{2}}=9.\] \[\therefore \,\,\,\,{{(k+3)}^{2}}+{{(h-3)}^{2}}=9\] So, required locus is \[{{(y+3)}^{2}}+{{(x-3)}^{2}}=9.\]