Question

If P is a point on the rectangular hyperbola x² – y² = a², C is its centre and S, S¢ are the two foci, then SP. S¢ P =

• A. 2
• B. (CP)²
• C. (CS)²
• D. (SS¢)²

Answer 1

Answer: (B)

(CP)²

Answer 2

Let the coordinates of P be (x, y)

The eccentricity of the hyperbola is $\sqrt{1 + \frac{a^{2}}{a^{2}}}$= $\sqrt{2}$

So the coordinates of the foci are S(a$\sqrt{2}$, 0) and S¢

(–a$\sqrt{2}$, 0).

Equation of the corresponding directrices are x = a/$\sqrt{2}$ and x = –a/$\sqrt{2}$.

By definition of the hyperbola

SP = e (distance of P from x = a/$\sqrt{2}$)

=$\sqrt{2}$| x – a /$\sqrt{2}$|

Similarly S¢P =$\sqrt{2}$|x + a/$\sqrt{2}$|

So that SP . S¢ .P = 2 |x² – a²/2| = 2x² – a² = x² + y²

= (CP)²

(Q P lies on the hyperbola x² – y² = a²).