Chords of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` are drawn through the positive end of the minor axis. Then prove that their midpoints lie on the ellipse.

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

C

Explanation

Let (h,k) be the mid-point of a chord passing through the positive end of the minor axis of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1.` then the equation of the chord is
`(hx)/(a^(2))+(ky)/(b^(2))-1=(h^(2))/(a^(2))+(k^(2))/(b^(2))-1` [ using T=S']
`implies(hx)/(a^(2))+(ky)/(b^(2))=(h^(2))/(a^(2))+(k^(2))/(b^(2))`
this passes through (0,b)
`therefore (k)/(b)=(h^(2))/(a^(2))+(k^(2))/(b^(2))`
hence , the locus of (h,k) is `(x^(2))/(a^(2))+(y^(2))/(b^(2))=(y)/(b)` , which is an ellipse .