Question

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

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

Answer 1

The correct option is B

$(CP{)}^2$

Let the coordinates of P be (x, y)
The coordinates 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 $S(a\sqrt{2},0)\text{and}{S}^{\prime }(-a\sqrt{2},0).$
Equation of the corresponding directrices are $x=\frac{a}{\sqrt{2}}$ and $x=\frac{-a}{\sqrt{2}}$.
By definition of the hyperbola
$SP=e\cdot (\text{distance of}P\text{from}x=\frac{a}{\sqrt{2}})=\sqrt{2}|x-\left(\frac{a}{\sqrt{2}}\right)|\text{Similarly}{S}^{\prime }P=\sqrt{2}|x+\left(\frac{a}{\sqrt{2}}\right)|\Rightarrow SP\cdot S^{\prime }P=2|x^2-\frac{a^2}{2}|=2x^2-a^2=x^2+y^2=(CP{)}^2$
$(\because$ $P$ lies on the hyperbola $x^2-y^2=a^2)$