Question

If $P(x_1,y_1)$ is a point on the hyperbola $x^2-y^2=a^2$, then $SP.S^{\prime }P=.......$

• 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 1

Answer: (D)

Given, equation of hyperbola
$x^2-y^2=a^2$...(i)
If $P\left(x_1,y_1\right)$ is a point on Eq. (i), then
$x_1^2-y_1^2=a^2$...(ii)
Now, $SP=e{x}_1-a$
and $S{P}^{\prime }=e{x}_1+a$
$\therefore SP\cdot S{P}^{\prime }=\left(e{x}_1-a\right)\left(e{x}_1+a\right)$
$=e^2\cdot x_1^2-a^2$
$=2x_1^2-a^2\left(\text{ since, for }{x}^2-y^2=a^2,e=\sqrt{2}\right)$
$=2x_1^2-\left(x_1^2-y_1^2\right)$[using Eq. (i)]
$=x_1^2+y_1^2$