Ellipse practice · Mixed difficulty

If the line $x+2 y+4=0$ cutting the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points whose eccentric angles are $30^{\circ}$ and $60^{\circ}$ subtends a right angle at the orgin, then its equation is

If the foci and vertices of an ellipse be $\left( { \pm 1,0} \right)$ and $\left( { \pm 2,0} \right)$ then the minor axis of the ellipse is

The locus of the point of intersection of the perpendicular tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ is

The foci of an ellipse are $( \pm 2,0)$ and its eccentricity is $\frac{1}{2}$ then the equation of ellipse is $\frac{x^2}{a^2}+\frac{y^2}{12}=1$. The value of '${a}$' is

Suppose $S$ and $S'$ are foci of the ellipse $\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1.$ If $P$ is a variable point on the ellipse and if $\Delta$ is area of the triangles $PSS'$, then the maximum value $\Delta$ is

An ellipse passes through the foci of the hyperbola, $9 x^{2}-4 y^{2}=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is $\frac{1}{2}$, then which of the following points does not lie on the ellipse?

The value of $a$ for the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b)$, if the extremities of the latusrectum of the ellipse having positive ordinate lies on the parabola $x^2=-2(y-2)$, is

The line $x=a t^2$ meets the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in the real points iff

The latusrectum of the ellipse $9{x^2} + 15{y^2} = 144$ is

For $0 < \theta < \pi / 2$, if the eccentricity of the hyperbola $x^2-y^2 \operatorname{cosec}^2 \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$, then the value of $\theta$ is:

The equation of the normal at the point $P\left( {2,3} \right)$ on the ellipse $9{x^2} + 16{y^2} = 180$ is

The eccentricity, foci and the length of the latus rectum of the ellipse $x^2+4 y^2+8 y-2 x+1=0$ are respectively equal to:

The eccentricity of the conic $4{x^2} + 16{y^2} - 24x - 32y = 1$ is

Tangent is drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $(3 \sqrt{3} \cos \theta, \sin \theta)$ where $\theta \in(0, \pi / 2)$. Then the value of $\theta$ such that sum of intercepts on axes made by the tangent is minimum, is

The ellipse $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinates axes. Another ellipse $E_2$ passing through the point $(0,4)$ circumscribing the rectangle $R$. The eccentricity of the ellipse $E_2$, is

If the coordinates of the vertices of an ellipse are $\left( { - 6,1} \right)$ and $\left( {4,1} \right)$ and the equation of a focal chord passing through the focus on the right side of the centre is $2x - y - 5 = 0.$ The equation of the ellipse is

An ellipse has foci $(3,1),(1,1)$ and it passes through point $(1,3)$. Its eccentricity is equal to

Locus of all such points so that sum of its distances from $(2,-3)$ and $(2,5)$ is always 10, is

From a point on the axis of $x$ common tangents are drawn to the parabola $y^2=4 x$ and the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$. If these tangents form an equilateral triangle with their chord of contact w.r.t parabola, then set of exhaustive values of $a$ is

The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latusrectum, is

If the line $l x+m y+n=0$ cuts the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points whose eccentric angles differ by $\pi / 2$, then

The equation of the chord of $\frac{x^2}{36}+\frac{y^2}{9}=1$ which is bisected at $(2,1)$, is

If polar of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is always touching the curve $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, then locus of pole is

The latus rectum of the ellipse $4 x^2+9 y^2-8 x-36 y+4=0$ is

The eccentricity of the ellipse $25{x^2} + 16{y^2} - 150x - 175 = 0$ is

If a number of ellipses is described having the same major axis but a variable minor axis, such that each one of the tangents at the ends of their latusrectum passes through the fixed point, then the point of concurrency is

If $x\cos \alpha + y\sin \alpha = p$ is a tangent to the ellipse, then

The equations of tangents to the ellipse $3{x^2} + 4{y^2} = 5$, which are inclined at ${30^0}$ to the $x - axis$, are

If $\frac{x}{a} + \frac{y}{b} = \sqrt 2$ touches the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ then the eccentric angle of the point of contact is equal to

The curve represented by $x = 3(\cos t + \sin t),y = 4(\cos t - \sin t)$ is