Chords of an ellipse are drawn through the positive end of the minor axis. Then their mid-point lies on

Let (x₁y₁) be the midpoint of the chord  Equation of the chord is ${\mathrm{S}}_1={\mathrm{S}}_{11}\Rightarrow \frac{{\mathrm{xx}}_1}{{\mathrm{a}}^2}+\frac{{\mathrm{yy}}_1}{{\mathrm{b}}^2}-\left(\frac{{\mathrm{x}}_1^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}_1^2}{{\mathrm{b}}^2}\right)=0$

Sub (0, b) then $\frac{{\mathrm{x}}_1^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}_1^2}{{\mathrm{b}}^2}-\frac{{\mathrm{y}}_1}{\mathrm{b}}=0$, 
 The locus of $\mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)\text{ is }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}-\frac{\mathrm{y}}{\mathrm{b}}=0$
It represents an ellipse