Question

question is in attachment

​

Answer 1

Answer is option A

Answer is attached to the file

thank you

Answer 2

Let a point P(x1, y1) be on the circle C1: x²+ y² = a².       --- (1)

Let the tangents to the Hyperbola H1: x² - y² = a²  ---- (2), from P be PQ and PR, touching H1 at Q(x2, y2) and R(x3, y3).

   So PQ:  x *x2 - y * y2 = a², and  PR : x * x3 - y * y3 = a²       --- (3)

 

As PQ and PQ pass through P,

       x1 * x2 - y1* y2 = a²  and   x1 * x3 - y1 * y3 = a²      --- (4)

Equation of QR - Chord of Contact containing Q & R - is clearly,

           x1 * x - y1 * y = a².    --- (5)

Midpoint of chord of contact QR is: S(α, β) = [ (x2+x3)/2, (y2 +y3)/2 ].

Adding two equations in (4), we get  x1 α - y1 β = a²    --- (6)

Equation of chord of contact QR of H1 with its mid point at S(α, β) is given by formula: 

                                 T    =   S1    --- (7)

            ie., x α - β y - a²  =  α² - β² - a²

           =>  x α - y β = α² - β².      --- (8)

Equations (5) and (8) represent the same Chord of contact QR:

    So   x1 / α = y1 / β = a²/(α² - β²)

     or   x1 = α a²/(α²+β²)   and   y1 = β a²/(α² - β²)      --- (9)

Substitute (9) in eq (1) to get :

         (a⁴ α² + a⁴ β² )/(α² - β²)² = a²

Replace S(α, β) by (x,y) to get the locus.

       =>  (x² + y²) a² = (x² - y²)²