If P and Q are two points whose coordinates are `(a t^2,2a t)a n d(a/(t^2),(2a)/t)` respectively and S is the point (a,0). Show that `1/(S P)+1/(s Q)` is independent of t.

Options

A. a
B. 4a
C. 2a
D. `(2)/(a)`

Answer

C

Explanation

We have,
`SP=sqrt(("at"^(2)-a^(2))+("at"-0)^(2))=(t^(2)+t)`
Replacing t by `-(1)/(t)` in SP, we get `SQ=(2)/(t^(2))(1+t^(2))`
`:. (1)/(SP)+(1)/(SQ)=(1)/(a(t^(2)+1))+(t^(2))/(a(t^(2)+1))`
`rArr (1)/(SP)+(1)/(SQ)=(1)/(a)`
`rArr (2SPSQ)/(SP+SQ)=2arArr HM` of SP and SQ in 2a