P is a variable points on the hyperbola `x^2/a^2-y^2/b^2=1` whose vertex is `A(a,0)` The locus of the middle points AP is

Options

A. `((2x-a)^(2))/a^(2)-(2y^(2))/b^(2)=1`
B. `((2x-a)^(2))/a^(2)-(4y^(2))/b^(2)=1`
C. `((2x-a)^(2))/a^(2)-(8y^(2))/b^(2)=1`
D. None of these

Answer

2

Explanation

Let the point P be `(a sectheta, b tantheta)`
and A(a, 0),
let the point of AP be (h, k)
`therefore h=(a+asectheta)/2 and k=(btantheta)/2`
`rArr(2h-a)/a=sectheta…(i)`
and `(2k)/b=tantheta…(ii)`
Squaring and Subtracting (ii) from (i) we get
`rArr((2h-a)^(2))/a^(2)-(4y^(2))/b^(2)=1`