Ellipse practice · Easy

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

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

$x = 4(1 + \cos \mathrm{\theta })$ 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

Let $LL'$ be the latusrectum and S be a focus of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1.$ If $\Delta SLL'$ is equilaterial, then the eccentricity of the ellipse is

Let the equations of two ellipses be

If the product of their eccentricities is $\frac{1}{2}$, then the length of theminor axis of ellipse $E_{2}$ is :

The foci of an ellipse are $\left( {0, \pm 4} \right)$ and the equations for the directrices are $y = \pm 9$. The equation for the ellipse is

The locus of midpoints of a focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

The acute angle between the pair of tangents drawn to the ellipse $2 x^2+3 y^2=5$ from the point $(1,3)$ is

The equation of the ellipse passing through (2,1) having $e = 1/2$ is

If pair of tangents drawn to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ from a point $P$, so that angle between the tangents are at right angle, then possible coordinates of the point $P$ is/are

The equation of the line passing through the centre and bisecting the chord $7 x+y-1=0$ of the ellipse $\frac{x^2}{1}+\frac{y^2}{7}=1$ is

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

If chords of contact of the tangent from two points $(x_1, y_1)$ and $(x_2, y_2)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are at right angles, then $\frac{x_1x_2}{a^2} \times \frac{y_1y_2}{b^2}$ is equal to

The sum of focal distance of any point on the ellipse with major and minor axes as $2a$ and $2b$ respectively, is equal to

The eccentricity of conic $4(2 y-x-3)^2+9(2 x+y-1)^2=$ 80 is

The parametric representation of a point of the ellipse whose foci are (3, 0) and $\left( { - 1,0} \right)$ and eccentricity $\frac{2}{3}$ is

Let ${E_1}:\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,\,\,a > b$. Let ${E_2}$ be another ellipse such that it touches the end points of major axis of ${E_1}$ and the foci of ${E_2}$ are the end points of minor axis of ${E_1}$. If ${E_1}$ and ${E_2}$ have same eccentricities, then its value is

If $P$ is a point on the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{20}} = 1$ whose foci are $S$ and $S'$. Then $PS + PS'$ is

Locus of the middle points of all chords of $\frac{x^2}{4}+\frac{y^2}{9}=1$, which are at a distance of 2 units from the vertex of parabola $y^2=-8ax$, is

If $\alpha-\beta=$ constant, then the locus of the point of intersection of tangents at $P(a \cos \alpha, b \sin \alpha)$ and $Q(a \cos \beta, b \sin \beta)$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, is

If $f(x)$ is a decreasing function, then the set of values of '$k$', for which the major axis of the ellipse $\frac{x^2}{f\left(k^2+2 k+5\right)}+\frac{y^2}{f(k+11)}=1$ is the $X$-axis, is

Let $P$ be the variable point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ with focii at $S \& S^{\prime}$. Then maximum area of $\triangle \mathrm{PSS}^{\prime}$