Basics of Ellipse

With respect to the ellipse $4x^2+7y^2=8$, the correct statement(s) is/are -

For the ellipse $9x^2+16y^2-18x+32y-119=0$, which of the following is/are true -

Eccentric angle of a point on the ellipse $x^2+3y^2=6$ at a distance $\sqrt{3}$ units from the centre of the ellipse is -

If latus rectum of an ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1 \{0<b<4\}$, subtend angle $2\theta$ at farthest vertex such that $cosec\theta =\sqrt{5}$, then -

If point $P(\alpha +1,\alpha )$ lies between the ellipse $16x^2+9y^2-16x=0$ and its auxiliary circle, then -

where [.] denotes greatest integer function.

If the chord through the points whose eccentric angles are $\theta \& \varphi$ on the ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$passes through the focus, then the value of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$ is -

If $\tan {\theta }_1·\tan {\theta }_2=-\frac{a^2}{{ b}^2}$ then the chord joining two points ${\theta }_1 \& {\theta }_2$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{{ b}^2}=1$ will subtend a right angle at

Point $\text{'}O\text{'}$ is the centre of the ellipse with major axis $AB$ and minor axis $CD$. Point $F$ is one focus of the ellipse. If $OF=6$ and the diameter of the inscribed circle of triangle $OCF$ is $2 ,$ then the product $\left(AB\right)\left(CD\right)$ is equal to -

The length of the normal (terminated by the major axis) at a point of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is -

where $r$ and $r_1$ are the focal distance of the point.

An ellipse is such that the length of the latus rectum is equal to the sum of the lengths of its semi principal axes. Then -

An ellipse is drawn with major and minor axes of lengths $10$ and $8$ respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is-

If the distance of a point on the ellipse $\frac{x^2}{6}+\frac{y^2}{2}=1$ from the centre is $2,$ then the eccentric angle is-

$Q$ is a point on the auxiliary circle of an ellipse. P is the corresponding point on ellipse. $N$ is the foot of perpendicular from focus $S$, to the tangent of auxiliary circle at $Q$, then -

If the normals at the points $P, Q, R$ with eccentric angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are concurrent, then show that $\left|\begin{array}{ccc}\mathrm{sin\alpha }& \mathrm{cos\alpha }& \sin 2\mathrm{\alpha }\\ \mathrm{sin\beta }& \mathrm{cos\beta }& \sin 2\mathrm{\beta }\\ \mathrm{sin\gamma }& \mathrm{cos\gamma }& \sin 2\mathrm{\gamma }\end{array}\right|=0$

If the normal at a point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ of semi axes $a, b$ centre $C$ cuts the major & minor axes at $G \& g$, show that $a^2·{\left(CG\right)}^2+b^2·{\left(Cg\right)}^2={\left(a^2-b^2\right)}^2$. Also prove that $CG=e^{2}CN$, where $PN$ is the ordinate of $P$. $(N$ is foot of perpendicular from $P$ on its major axis$)$

$ABC$ is an isosceles triangle with its base $BC$ twice its altitude. A point $P$ moves within the triangle such that the square of its distance from $BC$ is half the area of rectangle contained by its distances from the two sides. Show that the locus of $P$ is an ellipse with eccentricity $\sqrt{\frac{2}{3}}$ passing through $B\&C$.

The tangent at the point $\alpha$ on a standard ellipse meets the auxiliary circle in two points which subtends a right angle at the centre. Show that the eccentricity of the ellipse is ${\left(1+{\sin }^2\alpha \right)}^{-\frac{1}{2}}$.

Find the latus rectum, eccentricity, coordinates of the foci, coordinates of the vertices, the length of the axes and the centre of the ellipse $4x^2+9y^2-8x-36y+4=0$.

If set of value(s) of $\mathrm{\alpha }$ for which the point $\left(7-\frac{5}{4}\alpha ,\alpha \right)$ lies inside the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $\left(\frac{12}{\lambda },\frac{16}{\lambda }\right)$, then value of $\lambda$ is

An ellipse passes through the points $\left(-3,1\right)$ & $\left(2,-2\right)$ its principal axis are along the coordinate axes in order. The value of${\left(\mathrm{length} \mathrm{of} \mathrm{semi}-\mathrm{minor} \mathrm{axis}\right)}^2$  is equal to