If P is any point on hyperbola whose axis are equal,prove that SP S'P=CP2 - Maths - Conic Sections

Question

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

Aakash Sharma · Accepted Answer

Dear Student!

Here is the answer to your query.

For an hyperbole if the length of semi transverse and semi conjugate axes are equal.

Then a = b

∴ Equation of the given hyperbole is

x² – y² = a² ......(1)

$Then e = \sqrt{2}, C = (0, 0), S= (\sqrt{2} a, 0), S' = (- \sqrt{2} a, 0)$

Let coordinates of any point P on hyperbole be (α, β). Since P lies on (1)

∴ α² – β² = a² ......(2)

${Now SP}^{2} = {(\sqrt{2} a - α)}^{2} + β^{2} = 2 a^{2} + α^{2} + β^{2} - 2 \sqrt{2} a α a n d S' P^{2} = - (- \sqrt{2} a - α)^{2} + β^{2} = 2 a^{2} + α^{2} + β^{2} + 2 \sqrt{2} a α$

Now SP² .S'P² = (2a² + a² + β²)² – 8a²α²

= 4a⁴ + 4a² (α²+ β²) + (α²+ β²)² – 8a²α²

= 4a² (a²– 2α²) + 4a² (α²+ β²) + (α²+ β²)²

= 4a² (α²– β²– 2α²) + 4a² (α²+ β²) + (α²+ β²)²

= (α²+ β²)²= CP⁴

∴ SP. S'P = CP²

Cheers!

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

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