Find the locus of the middle points of all chords of `(x^2)/4+(y^2)/9=1` which are at a distance of 2 units from the vertex of parabola `y^2=-8a xdot`

Options

A. `(x^(2)/4 + y^(2)/9)^(2) = (xy)/(6)`
B. `(x^(2)/4 + y^(2)/9)^(2) = 4 (x^(2)/16 + y^(2)/81)`
C. `(x^(2)/4 + y^(2)/9)^(2) = x^(2)/9 + y^(2)/4`
D. none of these

Answer

B

Explanation

Let (h, k) be the mid-point of a chord of the ellipse `x^(2)/4 + y^(2)/9 = 1`. Then, its equation is
`(hx)/(4) + (ky)/(9) - 1 = h^(2)/4 + k^(2)/9 - 1 rArr (hx)/4 + (ky)/(9) = h^(2)/4 + k^(2)/9`
If is at a distance of 2 units from the vertex (0, 0) of the parabola `y^(2) = -8 ax`.
`therefore |(h^(2)/4 + k^(2)/9)/(sqrt(h^(2)/16 + k^(2)/81))| = 2 rArr (h^(2)/4 + k^(2)/9)^(2) = 4(h^(2)/16 + k^(2)/81)`
Hence, the locus of (h, k) is `(x^(2)/4 + y^(2)/9)^(2) = 4(x^(2)/16 + y^(2)/81).`