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Question

If P is any point on the hyperbola whose axis are equal, prove that SP. S'P = CP2.

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Solution

Equation of the hyperbola:
x2a2-y2b2=1

If the axes of the hyperbola are equal, then a=b.
Then, equation of the hyperbola becomes x2-y2=a2.

∴ b2=a2e2-1⇒a2=a2e2-1⇒1=e2-1⇒e2=2⇒e=2
Thus, the centre C0,0 and the focus are given by S2a,0 and S'-2a,0, respectively.
Let Pα,β be any point on the parabola.
So, it will satisfy the equation.
α2-β2=a2

∴ SP2=2a-α2+β2=2a2+α2-22aα+β2

S'P2=-2a-α2+β2=2a2+α2+22aα+β2

Now, SP2.S'P2=2a2+α2-22aα+β22a2+α2+22aα+β2=4a4+4a2α2+β2+α2+β22-8a2α2=4a2a2-2α2+4a2α2+β2+α2+β22=4a2α2-β2-2α2+4a2α2+β2+α2+β22=-4a2α2+β2+4a2α2+β2+α2+β22=α2+β22=CP4
∴ SP.S'P=CP2

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