S L Loney Solutions for Chapter: The Ellipse, Exercise 4: EXAMPLES XXXV

Prove that the locus of the intersection of normals at the ends of conjugate diameters of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is the curve
$2{\left(a^2{x}^2+b^2{y}^2\right)}^3={\left(a^2-b^2\right)}^2{\left(a^2{x}^2-b^2{y}^2\right)}^2.$

Circles of constant radius $c$ are drawn to pass through the ends of a variable diameter of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Prove that the locus of their centres is the curve $\left(x^2+y^2\right)\left(a^2{x}^2+b^2{y}^2+a^2{b}^2\right)=c^2\left(a^2{x}^2+b^2{y}^2\right)$.

Prove that the sum of the angles that the four normals drawn from any point to an ellipse make with the axis is equal to the sum of the angles that the two tangents from the same point make with the axis.

Triangles are formed by pairs of tangents drawn from any point on the ellipse $a^2{x}^2+b^2{y}^2={\left(a^2+b^2\right)}^2$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, and their chord of contact. Prove that the orthocenter of each such triangle lies on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

An ellipse is rotated through a right angle in its own plane about its center, which is fixed; prove that the locus of the point of intersection of a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in its original position, with the tangent at the same point of the curve in its new position is $\left(x^2+y^2\right)\left(x^2+y^2-a^2-b^2\right)=2\left(a^2-b^2\right)xy$.

If $Y$ and $Z$ are the feet of the perpendiculars from the foci upon the tangent at any point $P$ of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, prove that the tangents at $Y$ and $Z$ to the auxiliary circle, meet on the ordinates of $P$, and that the locus of their point of intersection is another ellipse.

Prove that the directrices of the two parabolas that can be drawn to have their foci at any given point $P$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and to pass through its foci meet at an angle which is equal to twice the eccentric angle of $P.$

The chords at right angles are drawn through any point $P$ on the ellipse, and the line joining their extremities meets the normal in the point $Q$. Prove that $Q$ is the same for all such chords, its coordinates being $\frac{a^3{e}^2\cos \alpha }{a^2+b^2}$ and $\frac{-a^{2}b{e}^2\sin \alpha }{a^2+b^2}$. Also, prove that the major axis is the bisector of the angle $PCQ$, and the locus of $Q$ for different positions of $P$ is the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}={\left(\frac{a^2-b^2}{a^2+b^2}\right)}^2$.