Discussion Forum



1)

 If   $P( x_{1},y_{1})$  is a point  on the hyperbola  x2-y2 =a2 , then SP.S'P=.....


A) $\frac{x_{1}^{2}-y_{1}^{2}}{a^{2}}$

B) $\frac{x_{1}^{2}+y_{1}^{2}}{a^{2}}$

C) $x_{1}^{2}-y_{1}^{2}$

D) $x_{1}^{2}+y_{1}^{2}$

Answer:

Option D

Explanation:

 Given , equation of hyperbola

    x2-y2 =a2 ..............(i)

 If P(x1,y1) is a point on Eq.(i) , then  

      $x_{1}^{2}-y_{1}^{2}$=$a_{}^{2}$  ......(ii)

 Now,     SP= ex1-a

     and SP'= ex1+a

 $\therefore$    SP.SP' = (ex1-a)(ex1 +a)

                         = e2.x12-a2

 = $2x_1^2-a^{2}$  (since , for x2-y2= a2, e = $\sqrt{2}$)

  = $2x_1^2-(x_1^2-y_1^2)$    [using Eq. (ii)]

 =  $x_1^2+y_1^2$