Ellipse practice · Mixed difficulty

In the ellipse, the minor axis is 8 and eccentricity is $\sqrt{5} / 3$. Then major axis is

If $\alpha$ and $\beta$ are eccentric angles of the ends of $a$ focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$ is equal to

The normal at the point $(3,2 \sqrt{3})$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ meets the major axis of the ellipse at $(\alpha, 0)$, the value of $\alpha$ is:

Tangent are drawn from any point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to the circle $x^2+y^2=r^2$, then the chord of contact is a tangent to the ellipse, is

In an ellipse length of minor axis is 8 and eccentricity is $\frac{{\sqrt 5 }}{3}$. The length of major axis is

Locus of the point which divides double ordinate of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ in the ratio $1:2$ internally, is

Three points $A,B,C$ are taken on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with eccentric angles $\mathrm{\theta },\mathrm{\theta } + \mathrm{\alpha }\,\mathrm{and }\,\mathrm{\theta } + 2\mathrm{\alpha }$, then

Equation of the latusrectum of the ellipse $9{x^2} + 4{y^2} - 18x - 8y - 23 = 0\;$ are

The equation of ellipse, whose distance between the foci is equal to 8 and the distance between the directrices is 18 , is

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms. Are

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ be a given ellipse, length of whose latusrectum is 10 . If its eccentricity is the maximum value of the function $\phi(t)=\frac{5}{12}+t-t^2$, then $a^2+b^2$ is equal to

The vertices of the ellipse $9 x^2+4 y^2-18 x-27=0$, are

The product of the perpendiculars from the foci of the ellipse $\frac{x^2}{144}+\frac{y^2}{100}=1$ on any tangent is:

A focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$, then length of semi major axis is

All ellipse has its centre at $\left( {1, - 1} \right)$ and semi-major axis $= 8$ and it passes through the point $\left( {1,3} \right)$. The equation of the ellipse is

If a triangle is inscribed in an ellipse and two of its sides are parallel to given straight lines, then envelope of the third side is

The equation of the ellipse whose foci are $\left( { \pm \,2,0} \right)$ and eccentricity $\frac{1}{2}$ is

Equation of the tangent to the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ perpendicular to the line $2 x+y+7=0$, is

Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$. The coordinates of $A$ and $B$, are

The eccentricity of the ellipse $9 x^2+25 y^2=225$ is

Let $\mathrm{E}$ be the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and $C$ be the circle $x^2+y^2=9$. Let $P$ and $Q$ be the points $(1,2)$ and $(2,1)$, respectively. Then

$\frac{(3 x-4 y+10)^2}{2}+\frac{(4 x+3 y-15)^2}{3}=1$ is an ellipse, then find major & minor axes

The tangent at the point $P\left(\frac{4}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right)$ to the ellipse $9 x^2+16 y^2=144$ meets the axis of $x$ at $A$ and the axis of $y$ at $B$. If $C$ is the centre of the ellipse, then area of the $\Delta A B C$ is (in sq. units)

$x = 4(1 + \cos \mathrm{\theta })$ and $y = 3(1 + \sin \mathrm{\theta })$ are the parametric equations of

$\mathrm{S}$ and $\mathrm{T}$ are the foci of the ellipse $\mathrm{x}^2 / \mathrm{a}^2+\mathrm{y}^2 / \mathrm{b}^2=1$ and $\mathrm{B}$ is an end of the minor axis. If $\mathrm{S T B}$ is an equilateral triangle, the eccentricity of the ellipse is $\mathrm{e}$ then find value of $4 \mathrm{e}$

The maximum value of $|x+y|$, where $(x, y)$ lies on the ellipse $4 x^2+y^2=4$ is

The equations to the directrices of the ellipse $4{\left( {x - 3} \right)^2} + 9{\left( {y + 2} \right)^2} = 144$ are

Which of the following conic has tangent $x+\sqrt{3} y-2 \sqrt{3}=0$ at point $\left(\frac{3 \sqrt{3}}{2}, \frac{1}{2}\right)$ ?

A circle has the same centre as an ellipse & passes through the focii $F_1$ & $F_2$ of the ellipse, such that the two curves intersect at 4 points. Let ' $P$ ' be any one of their points of intersection. If the major axis of the ellipse is $17$ & the area of the triangle $P F_1 F_2$ is 30 , then the distance between the focii is :

A point $P$ moves so that sum of its distances from $\left( {-ae,0} \right)$ and $\left( {ae,0} \right)$ is $2a$. Then, the locus of $P$ is