The locus of the middle points of the chord of the ellipse $x^2/a^2+y^2/b^2=1$ touching  the ellipse  $x^2/{\alpha }^2+y^2/{\beta }^2=1$ is 

Equation of chord having $\left(x_1,y_1\right)\text{ as mid point is }x{x}_1/a^2+y{y}_1/b^2=x_1^2/a^2+y_1^2/b^2$

since it touches the ellipse $\frac{x^2}{{\alpha }^2}+\frac{y^2}{{\beta }^2}=1\text{ then }{\alpha }^2\frac{x_1^2}{a^4}+{\beta }^2\cdot \frac{y_1^2}{b^4}={\left(\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}\right)}^2$

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