Question

If P is a point on the rectangular hyperbola $x^2-y^2=a^2$, C is its centre and S, S' are the two foci, then $SP.S^{\prime }P=$

• A. $2$
• B. ${\left(CP\right)}^2$
• C. ${\left(CS\right)}^2$
• D. ${\left(S{S}^{\prime }\right)}^2$

Answer 1

Answer: (B)

${\left(CP\right)}^2$

Let the coordinates of $P$ be $(x,y)$
The co-ordinates of the centre C are $(0,0)$
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 $s^{\prime }(-a\sqrt{2},0)$.
Equation of the corresponding directrices are $x=a/\sqrt{2}$ and $S=-a\sqrt{2}$.
By definition of the hyperbola
SP$=e$(distance of $P$ from $x=a/\sqrt{2}$)
$=\sqrt{2}|x-\left(a/\sqrt{2}\right)|$
Similarly $S^{\prime }P=\sqrt{2}|x+\left(a/\sqrt{2}\right)|$
So that $SP.S^{\prime }P=2|x^2-\left(a^2/2\right)|=2x^2-a^2$
$=x^2+y^2={\left(CP\right)}^2$ ($\because$ P lies on the hyperbola $x^2-y^2=a^2$)
Hence, option 'B' is correct.