Dr. SK Goyal Solutions for Chapter: Ellipse, Exercise 4: EXERCISE ON LEVEL-I

The tangent and normal to the ellipse $x^2+4y^2=4$ at a point $P\left(\theta \right)$ on it meets the major axis in $Q$ and $R$ respectively. If $QR=2$, show that the eccentric angle '$\theta$' of $P$ is given by $\cos \theta =\pm \left(\frac{2}{3}\right)$.

A circle passes through the end of a diameter of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and also touches the curve. Prove that the locus of its centre is the ellipse $4\left(a^2{x}^2+b^2{y}^2\right)=\left(a^2-b^2\right).$

$Q$ is the point on the auxiliary circle corresponding to the point $P$ on the ellipse. $PLM$ is drawn parallel to the radius $CQ$ to meet the axes in $L$ and $M$. Prove that $PM$ and $PL$ are equal to the semi-axes.

If $\alpha +\beta =\gamma$ (a constant), prove that tangents at '$\alpha$' and '$\beta$' on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, intersect on the diameter through '$\gamma$'.

If $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the tangents at which meet in $\left(h,k\right)$ and the normals in $\left(p,q\right)$. Prove that $a^{2}p=e^{2}h{x}_1{x}_2$ and $b^{4}q=e^{2}k{y}_1{y}_2{a}^2$ where $e$ is the eccentricity.

Prove that the equation of the locus of the point of intersection of the tangent at one end of a focal chord of an ellipse with the normal at the other end is $\frac{x^2}{a^2}+\frac{b^2{y}^2}{{\left(2a^2-b^2\right)}^2}=1.$

If $PSQ$ and $PHR$ be two chords of an ellipse through its foci $S$ and $H$, then prove that $\frac{PS}{SQ}+\frac{PH}{RH}=\frac{2\left(1+e^2\right)}{\left(1-e^2\right)}$ where $e$ is the eccentricity of ellipse.

An ellipse of semi axes $a,b$ slides between two perpendicular lines, prove that the locus of its foci is $\left(x^2+y^2\right)\left(x^2{y}^2+b^4\right)=4a^2{x}^2{y}^2$, the two lines being taken as the axes of co-ordinates.