Ellipse practice · Easy

If a bar of given length moves with its extremities on two fixed straight lines at right angles, then the locus of any point on bar marked on the bar describes a/an

The locus of the middle point of the portion of a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ included between the axes is the curve

If $\left( { - 4,3} \right)$ and $\left( {8,3} \right)$ are the vertices of an ellipse whose eccentricity is $5/6$ then the equation of the ellipse is

Length of a latera recta of the ellipse $\frac{x^2}{81}+\frac{y^2}{63}=1$ is (in units)

If the normal at the point $P(\theta)$ to the ellipse $\frac{x^2}{14}+\frac{y^2}{5}=1$ intersects it again at the point $Q(2 \theta)$, then $\cos \theta$ is equal to

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

The equation of the ellipse whose axes are of length 6 and $2 \sqrt{6}$ and their equations are $x-3 y+3=0$ and $3 x+y-1=0$ respectively, is

The eccentricity of the conic $\frac{{{{\left( {x + 2} \right)}^2}}}{7} + {\left( {y - 1} \right)^2} = 14$ is

$AB$ is a diameter of ${x^2} + 9{y^2} = 25.$ The eccentric angle of $A$ is $\pi /6$. Then, the eccentric angle of $B$ is

The equation of a directrix of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1$ is

Consider an ellipse with major and minor axes of length 10 and 8 units respectively. The radius of largest circle that can be inscribed in this ellipse, it is given that centre of this circle is one focus of the ellipse is

The equation of chord of ellipse $2x^2+5y^2=20$ which is bisected at the point $(2,1)$

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latusrectum is

The equation of normal at the point $(0,3)$ of the ellipse $9 x^2+5 y^2=45$ is

If the distance between the directrices be thrice the distance between the foci, then the eccentricity of ellipse is

If the coordinates of two points $A\,\,\,\mathrm{and }\,\,B$ are $\left( {\sqrt {7,} 0} \right)$ and $\left( { - \sqrt {7,}0 } \right)$ respectively and $P$ is any point on the conic, $9{x^2} + 16{y^2} = 144,$ then $PA + PB$ is equal to

Let $S(3,4)$ and $S^{\prime}(9,12)$ be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus $S$ to a tangent to the ellipse is $(1,-4)$, then the eccentricity of the ellipse, is

The position of the point $(1,3)$ w.r.t. the ellipse $4 x^2+9 y^2-16 x$ $-54 y+61=0$ is

Eccentricity of the ellipse ${x^2} + 2{y^2} - 2x + 3y + 2 = 0$ is

If the line $y=m x+c$ touches the ellipse $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$, then find the value of $c$.

Equation of the tangents to the ellipse $\frac{x^2}{9}+\frac{y^2}{16}=1$ which are parallel to the line $x+y+1=0$, is

The curve represented by $x=2(\cos t+\sin t)$, $y=5(\cos t-\sin t)$, is

Let $S$ and $S^{\prime}$ be two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. If a circle described on $SS'$ as diameter intersects the ellipse in real and distinct points, then the eccentricity $e$ of the ellipse satisfies

The distances from the foci of point $P\left( {{x_1},{y_1}} \right)$ on the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{25}} = 1$ are

If two tangents drawn to the ellipse $x^2+4 y^2=4$ intersect perpendicularly at $P$, then the locus of $P$ is

Let $P$ be a point in the first quadrant lying on the ellipse $9 x^2+16 y^2=144$, such that the tangent at $P$ to the ellipse is inclined at an angle of $135^{\circ}$ to the positive direction of $x$-axis. Then the coordinates of $P$ are

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

The distance of the point $(\sqrt{6} \cos \theta, \sqrt{2} \sin \theta)$ on the ellipse from the centre of the ellipse is 2 if $\theta=$

If the coordinates of the centre, a focus and adjacent vertex are $\left( {2, - 3} \right),\left( {3, - 3} \right)$ and $\left( {4, - 3} \right)$ respectively, then the equation of the ellipse is

$x-2 y+4=0$ is a common tangent to $y^2=4 x$ and $\frac{x^2}{4}+\frac{y^2}{b^2}=1$. Then, the value of $b$ and the other common tangent are given by