If `P(x_(1),y_(1))` is a point on the hyperbola `x^(2)-y^(2)=a^(2)`, then `SO.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`
and `SP'=ex_(1)+a`
`thereforeSPdotSP'=(ex_(1)-a)(ex_(1)+a)`
`=e^(2)dotx_(1)^(2)-a^(2)`
`=2x_(1)^(2)-a^(2)` (since, for `x^(2)-y^(2)=a^(2),e=sqrt2`)
`=2x_(1)^(2)-(x_(1)^(2)-y_(1)^(2))` [using Eq. (i)]
`=x_(1)^(2)+y_(1)^(2)`