Ellipse practice · Medium

Length of major axis of ellipse $9{x^2} + 7{y^2} = 63$ is

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

Equation of a tangent to the ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$, passing through the point where a directrix of the ellipse meets the + ve $x$-axis is

An ellipse passing through the point $(2 \sqrt{13}, 4)$ has its foci at $(-4,1)$ and $(4,1).$ Then its eccentricity is

The locus of the foot of the perpendicular drawn from the centre to any tangent to the ellipse $x^2 / a^2+y^2 / b^2=1$ is

Let $S$,$S'$ be the foci and $BB'$ be the minor axis of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1.$ If $\angle BSS' = \theta ,$ then the eccentricity $e$ of the ellipse is equal to

If $P\left( \theta \right)$ and $Q\left( {\pi /2 + \theta } \right)$ are two points on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1.$ Locus of the mid-point of $PQ$ is

If the tangent at $P(\theta)$ on the ellipse $16x^2 + 11y^2 = 256$ touches the circle $x^2 + y^2 + 2x – 15 = 0,$ then $\theta =$

From a point P, perpendicular tangents $P Q$ and $P R$ are drawn to ellipse $x^2+4 y^2=4$. The locus of the circumcentre of $\Delta P Q R$, is

Let $y=2 x$ be one of the diagonal of a parallelogram inscribed in an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 ; a>b$ having eccentricity $\frac{\sqrt{a^2-8}}{a}$. If the maximum area of parallelogram is $16 \sqrt{2}$, then 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 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:

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

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

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 $P_{1}$ and $P_{2}$ are two points on the ellipse $\frac{x^{2}}{4}+y^{2}=1$ at which the tangents are parallel to the chord joining the points $(0,1)$ and $(2,0)$, then the distance between $P_{1}$ and $P_{2}$ is

In the normal at the end of latusrectum of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with eccentricity $e$, passes through one end of the minor axis, then

If $A(1,4)$ and $B(3,0)$ then the maximum area of the triangle $\triangle A B C$ inscribed inscribed in ellipse $2 x^2+y^2=18$ is

The normal at $\left(2, \frac{3}{2}\right)$ to the ellipse, $\frac{x^{2}}{16}+\frac{y^{2}}{3}=1$ touches a parabola, whose equation is

If ${M_1}$ and ${M_2}$ are the feet of the perpendiculars from the foci ${S_1}$ and ${S_2}$ of the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{16}} = 1$ on the tangent at any point $P$ on the ellipse, then $\left( {{S_1}{M_1}} \right)\left( {{S_2}{M_2}} \right)$ is equal to

The locus of the point of the chord of contact of tangents from which the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches the circle, is

If tangents are drawn to the ellipse $x^2+2 y^2=2$ then locus of mid-points of intercepts made by the tangents between the coordinate axes is

If normal to ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at $\left(a e, \frac{b^2}{a}\right)$ is passing through $(0,-2 b)$, then value of eccentricity is

If $F_1$ and $F_2$ are the foci of the ellipse $4 x^2+9 y^2=36$, $P$ is a point on the ellipse such that $P F_1: P F_2=$ $2: 1$; then area of $\Delta P F_1 F_2$ is:

The equation of the ellipse whose axes are along the co-ordinate axes, vertices are $( \pm 5,0)$ and foci at $( \pm 4,0)$, is $\frac{x^2}{25}+\frac{y^2}{b^2}=1$. The value of ${b}^2$ is

$\frac{x^2}{f(4 a)}+\frac{y^2}{f\left(a^2-5\right)}=1$ represents an ellipse with major axis as $y-$ axis and ' $f$ ' is decreasing function, positive for all ' $a$ ', then a belongs to

Consider the particle travelling clockwise on the elliptical path $\frac{x^2}{100}+\frac{y^2}{25}=1$. The particle leaves the orbit at the point $(-8,3)$ and travels in a straight line tangent to the ellipse, then the point will the particle cross the $Y$-axis is

Let $P, Q, R$ be three points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $P^{\prime}, Q^{\prime}$ and $R^{\prime}$ be the corresponding points on the auxiliary circle. Then,
Area of $\triangle P Q R$ : Area of $\triangle P^{\prime} Q^{\prime} R^{\prime}=$