Question

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

Answer 1

Given: Axis of the hyperbola are equal, i.e. a = b 

To prove: SP.S’P = CP² 

Formula used: 

The standard form of the equation of the hyperbola is,

Foci of the hyperbola are given by (±ae, 0)

\(\Rightarrow\) Foci of hyperbola are given by \(\big(\pm \sqrt{2a},0\big)\)

So, S \(\big(\sqrt{2a},0\big)\) and S'\(\big(-\sqrt{2a},0\big)\)

Let P (m, n) be any point on the hyperbola

The distance between two points (m, n) and (a, b) is given by \(\sqrt{(m-a)^2 + (n - b)^2}\)

SP = \(\sqrt{(m-\sqrt{2}a)^2 + (n - 0)^2}\)

\(\Rightarrow\) SP² = m² + 2a² - 2\(\sqrt{2}\) am + n²

S'P = \(\sqrt{(m-\sqrt{2}a)^2 + (n - 0)^2}\)

 \(\Rightarrow\) S'P² = m² + 2a² - 2\(\sqrt{2}\) am + n²

C is Centre with coordinates (0, 0)

Now,

From (i):

\(\Rightarrow\) SP². S'P² = CP⁴

Taking square root both sides:

Hence Proved