Ellipse practice · Easy

The foci of the ellipse $\frac{(x-3)^2}{36}+\frac{(y+2)^2}{16}=1$, are

Length of the major axis of the ellipse $9 x^2+7 y^2=63$, 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

Let the eccentricity of an ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,\,a > b,$ be $\frac{1}{4}.$ If this ellipse passes through the point $\left( { - 4\sqrt {\frac{2}{5}} ,3} \right),$ then ${a^2} + {b^2}$ is equal to:

If the length of the major axis of an ellipse is $\frac{{17}}{8}$ times the length of the minor axis, then the eccentricity of the ellipse is

The locus of the point of intersection of tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the points whose eccentric angles differ by $\pi / 2,$ is

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

If the eccentricity of the ellipse $\frac{x^2}{a^2+2}+\frac{y^2}{a^2+5}=1$ be $\frac{1}{\sqrt{3}}$, then length of latus rectum of ellipse is

The equation $14 x^2-4 x y+11 y^2-44 x-58 y+71=0$ represents

Consider ellipses $E_k: k x^2+k^2 y^2=1, k=1,2, \ldots, 20$.

Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_k$. If $r_k$ is the radius of the circle $C_k$, then the value of $\sum\limits_{{k=1}}^{20} \frac{1}{r_k^2}$ is

The equation of the tangent at the point $\left(\frac{1}{4}, \frac{1}{4}\right)$ of the ellipse $\frac{x^2}{4}+\frac{y^2}{12}=1$ is

The eccentricity of an ellipse is $2 / 3$, its latus rectum is 5 and its centre is $(0,0)$. The equation of the ellipse is

Let $x=4$ be a directrix to an ellipse whose centre is at the origin and eccentricity is $\frac{1}{2}$.If $P(1, \beta), \beta>0$ is a point on the ellipse, then the equation of the normal to it at $P$, is

The equation of the tangents to the ellipse $4 x^2+3 y^2=5$ which are parallel to the line $y=3 x+7$ are

Equation of the normal to the ellipse $4{\left( {x - 1} \right)^2} + 9{\left( {y - 2} \right)^2} = 36$, which is parallel to the line $3x - y = 1$, is

Let $x = 4$ be a directrix to an ellipse whose centre is at origin and its eccentricity is $\frac{1}{2}\,.$ If $P\left( {1,\,\beta } \right),\,\,\,\beta > 0\,$ is a point on this ellipse, then the equation of the normal to it at $P$ is

Given $f$ is increasing, the equation $\frac{x^2}{f(2 a)}+\frac{y^2}{f\left(a^2-3\right)}=1$ represents an ellipse with $X$-axis as major axis if

A point on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ at a distance equal to the mean of lengths of the semi-major and semi-minor axis from the centre, is

Length of latusrectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, is

The curve with parametric equations $x = \mathrm{\alpha } + 5\cos \mathrm{\theta },\;y = \mathrm{\beta } + 4\sin \mathrm{\theta }$ (where $\theta$ is parameter) is

The eccentricity of the hyperbola whose length of the latusrectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is

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

If the eccentricity of the hyperbola $x^2-y^2 \operatorname{coces}^2 \alpha=25$ is $\sqrt{5}$ times the eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \alpha+y^2=5$, then $\alpha$ is equal to :

The eccentricity of an ellipse $\frac{x^{2}}{a^2}+\frac{y^2}{b^2}=1$ whose latusrectum is half of its minor axis is

For an ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ with vertices $A$ and $A^{\prime}$, tangent drawn at the point $P$ in the first quadrant meets the $y$-axis in $\mathrm{Q}$ and the chord $\mathrm{A}^{\prime} \mathrm{P}$ meets the $\mathrm{y}$-axis in $\mathrm{M}$. If ' $\mathrm{O}$ ' is the origin then $\mathrm{OQ}^2-\mathrm{MQ}^2$ equals to

The eccentricity of the ellipse $9{x^2} + 5{y^2} - 18x - 20y - 16 = 0$ is

If the point $p$ on the ellipse $4 x^2+5 y^2=20$ is farthest from the point $Q(0,-4)$, then $P Q^2$ is equal to

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, where mid-point is $\left(\frac{1}{2}, \frac{2}{3}\right)$ is

The eccentricity of an ellipse, with centre at the origin is $2/3.$ If one of directrices is $x = 6,$ then equation of the ellipse is

The curve represented by the equation $4{x^2} + 16{y^2} - 24x - 32y - 12 = 0$ is