Ellipse practice · Mixed difficulty

The point, at shortest distance from the line $x + y = 7$ and lying on an ellipse ${x^2} + 2{y^2} = 6$, has coordinates

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is?

The equation of the circle drawn with the two foci of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as the end-points of a diameter, is

In the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the equation of diameter conjugate to the diameter $y=\frac{b}{a} x$ is

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

The sum of the squares of the perpendiculars on any tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ from two points on the minor axis each at a distance $\sqrt {{a^2} - {b^2}}$ from the centre 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:

The eccentricity of the ellipse, whose latus rectum is equal to the distance between two focus points, is

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

If $p$ is the length of the perpendicular from a facus upon the tangent at any point $p$ of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and $r$ is the distance of $p$ from the focus, then $\frac{{2a}}{r} - \frac{{{b^2}}}{{{p^2}}}$ is equal to

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

Let $E$ be the ellipse $\frac{{{x^2}}}{8} + \frac{{{y^2}}}{4} = 1$ and $C$ be the circle ${x^2} + {y^2} = 9$. Let $P$ and $Q$ be the points $\left( {1,2} \right)$ and $\left( {2,1} \right)$ respectively. Then

The equation of a diameter conjugate to a diameter $y = \frac{b}{a}x$ of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ is

The locus of the point of intersection of tangents to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ which meet at right angle is

In an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on

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

If $C$ is the centre of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and the normal at an end of a latusrectum cuts the major axis in $G,$ then $CG =$

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

The locus of the middle points of the portions of the tangents of the ellipse $x^2 / a^2+y^2 / b^2=1$ included between the axis is the curve.

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

Tangents are drawn from the point on the line $x - y - 5 = 0$ to ${x^2} + 4{y^2} = 4$, then all the chords of contact pass through a fixed point, whose coordinates are