Q.Find the equation of the locus of a point the tangents from which to the ellipse $3x^2 + 5y^2 = 15$ are at right angles.

Two circles are naturally associated with every ellipse. The auxiliary circle, $x^2+y^2=a^2$, is drawn with the major axis as its diameter and is used purely as a geometric construction device for the eccentric-angle parametrisation — every point on the ellipse corresponds to a point on this circle sharing the same eccentric angle.

The director circle, $x^2+y^2=a^2+b^2$, is a genuinely different object: it is the locus of points in the plane from which the two tangents drawn to the ellipse are mutually perpendicular. It is derived by setting the product of the two tangent slopes, $m_1m_2=\frac{y_1^2-b^2}{x_1^2-a^2}$, equal to $-1$ (the condition for perpendicularity) and simplifying. Since $a^2+b^2>0$ always, the ellipse's director circle is always real and always strictly larger than its auxiliary circle. …