If `P(x_(1),y_(1))` is a point on the hyperbola `x^(2)-y^(2)=a^(2)`, then `SP.S'P=` . . . .

Options

A. `(x_(1)^(2)-y_(1)^(2))/(a^(2))`
B. `(x_(1)^(2)-y_(1)^(2))/(a^(2))`
C. `x_(1)^(2)-y_(1)^(2)`
D. `x_(1)^(2)-y_(1)^(2)`

Answer

D

Explanation

Given, equation of hyperbola
`x^(2)-y^(2)=a^(2)` . . (i)
If `P(x_(1),y_(1))` is a point on Eq. (i), then
`x_(1)^(2)-y_(1)^(2)=a^(2)` . . (ii)
Now, `SP=ex_(1)-a`
andd `SP=ex_(1)+a`
`thereforeSP*SP'=(ex_(1)-a)(ex_(1)+a)=e^(2)*x_(1)^(2)_a^(2)`
`=2x_(1)^(2)-a^(2)` (since, for `x^(2)-y^(2)=a^(2),e=sqrt(2)`)
`=2x_(1)^(2)(x_(1)^(2)-y_(1)^(2))` [using eq. (i)]
`=x_(1)^(2)+y_(1)^(2)`