The locus of the mid-points of the chords of the ellipse $\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)=1$ which are tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{{\beta }^2}=1$ is

Let the point be $(h, k)$.

The equation of the chord of the ellipse whose middle point $(h, k)$ is

$\frac{hx}{a^2}+\frac{ky}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$     …(i)

and the equation of the tangent to the ellipse

$\frac{x^2}{{\alpha }^2}+\frac{y^2}{{\beta }^2}=1$ at $(\alpha \cos \theta , \beta \sin \theta )$ is

$\frac{x}{\alpha }\cos \theta +\frac{y}{\beta }\sin \theta =1$

Since both the equations are identical, so

$\frac{\cos \theta /\alpha }{h/a^2}=\frac{\sin \theta /\beta }{k/b^2}=\frac{1}{\left(\frac{h^2}{a^2}+\frac{k^2}{b^2}\right)}$

Squaring and adding, we get

$\frac{{\alpha }^2{h}^2}{a^4}+\frac{{\beta }^2{k}^2}{b^4}={\left(\frac{h^2}{a^2}+\frac{k^2}{b^2}\right)}^2$

Hence the locus of $(h, k)$ is

 $\frac{{\alpha }^2{x}^2}{a^4}+\frac{{\beta }^2{y}^2}{b^4}={\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)}^2$