The locus of mid-points of focal chords of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` is

Options

A. `(x^(2))/(a^(2))+(y^(2))/(b^(2))=(ex)/(a)`
B. `(x^(2))/(a^(2))-(y^(2))/(b^(2))=(ex)/(a)`
C. `x^(2)+y^(2)=a^(2)+b^(2)`
D. none of these

Answer

A

Explanation

Let P (h,k) be the mid-point of a focal chord of the ellipse , then its equation is
`(hx)/(a^(2))+(ky)/(b^(2))=(h^(2))/(a^(2))+(k^(2))/(b^(2))`
It passes through (ae,0)
` therefore (h)/(a)e=(h^(2))/(a^(2))+(k^(2))/(b^(2))`
hence , the locus of (h,k) is `(x^(2))/(a^(2))+(y^(2))/(b^(2))=(xe)/(a)`