The locus of the image of the focus of the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{9}=1,(\mathrm{a}>\mathrm{b}),$ with respect to any of the tangents to the ellipse is

Let S'(h, k)be the image. SS' cuts a tangent at a point which lies on the auxiliary circle of the ellipse. Therefore,

${\left(\frac{\mathrm{h}\pm 4}{2}\right)}^2+\frac{{\mathrm{k}}^2}{4}=25$
Hence, the locus is $(\mathrm{x}\pm 4{)}^2+{\mathrm{y}}^2=100$.