Question

P and Q are two points whose coordinate are (at² , 2at), `(a/t^2 , (-2a)/t)` and S is the point (a, 0). Prove that `(1)/"SP" + (1)/"SQ"` is constant for all values of it.

Answer 1

The given points are P(at² , 2at), Q`(a/t^2 , (-2a)/t)` and S(a, 0)
Now, 
SP = `sqrt((at^2 - a)^2 + (2at - 0)^2)`
= `sqrt(a^2(t^4 + 1 - 2t^2) + 4a^2t^2)`
= `asqrt(t^4 + 1 + 2t^2)`
= `asqrt((t^2 + 1)^2`
SP = a(t² + 1)units.
Also, SQ = `sqrt((a/t^2 - a)^2 + ((-2a)/t - 0)^2)`
= `sqrt(a^2(1/t^4 + 1 - 2/t^2) + (4a^2)/t^2)`
= `asqrt(1/t^2 + 1 + 2/t^2)`
= `asqrt((1/t^2 + 1)^2`