Ellipse practice · Medium

Suppose $x$ and $y$ are real numbers and that $x^2+9 y^2-4 x+6 y+4=0$, then the maximum value of $(4 x-9 y)$.

If the length of the latusrectum of the ellipse ${x^2}{\tan ^2}\theta + {y^2}{\sec ^2}\theta = 1$ is $\frac{1}{2}$, then $\theta =$

If $Q$ is the point on the auxiliary circle corresponding to a point $P$ on an ellipse.
Then, the normals at $P$ and $Q$ meet on

A tangent at any point to the ellipse $4{x^2} + 9{y^2} = 36$ is cut by the tangent at the extremities of the major axis at $T$ and $T'$. The circle on $TT'$ as diameter passes through the point

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

Let the tangent and normal at the point $(3 \sqrt{3}, 1)$ on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let the circle $C$ be drawn taking $A B$ as a diameter and the line $x=2 \sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha, \beta)$, then $\alpha^{2}-\beta^{2}$ is equal to

If the normals at the points $A, B, C$ and $D$ of an ellipse meet in a point $O$, then $S A \cdot S B \cdot S C \cdot S D$ is equal to (where, $S$ is one of the foci)

If $lx + my + n = 0$ represents a chord of the ellipse ${b^2}{x^2} + {a^2}{y^2} = {a^2}{b^2}$ whose eccentric angles differ by ${90^0}$, then

The eccentricity of the conic $4 x^2+16 y^2-24 x-32 y=1$ is

An ellipse whose focii are $(2,4)$ and $(14,9)$ and touches x-axis, then its eccentricity is

Coordinates of the foci of the ellipse $2 x^2+3 y^2-4 x-12 y+13=0$ are

If the normal at any point $P$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ meets the major and minor axes at ${G}$ and $G^{\prime}$ respectively and ${OF}$ is the perpendicular drawn from centre ${O}$ to this normal, then $PF \cdot PG$ is equal to

If the tangents are drawn to the ellipse ${x^2} + 2{y^2} = 2$, then the locus of the mid point of the intercept made by the tangents between the coordinate axes is

If the line $x\cos \mathrm{\alpha } + y\sin \mathrm{\alpha } = p$ be normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, then

Let the ellipse $E: x^2+9 y^2=9$ intersect the positive $x$ - and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A, P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m-n$ is equal to

If $m$ is the slope of a common tangent to the curves $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1$ and $\,{{\text{x}}^2} + {y^2} = 12,$ then ${\text{12}}{m^2}$ is equal to:

The distance of the point '$\theta$' on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ from a focus is

The locus of the foot of the perpendicular from the foci on any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, is

The equation of the normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the positive end of the latusrectum, is

The equation of the tangents which are drawn at the ends of the major axis of the ellipse $9 x^2+5 y^2-30 y=0$ are

If $y = mx - \frac{{\left( {{a^2} - {b^2}} \right)m}}{{\sqrt {{a^2} + {b^2}{m^2}} }}$ is normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ for all values of $m$ belonging to

Ellipse $2 x^2+y^2=1$ and the line $y=m x$ meet at a point $P$ in the first quadrant. Normal to the ellipse at $P$ meets $x$-axis at $\left(-\frac{1}{3 \sqrt{2}}, 0\right)$ and $y$-axis at $(0, \beta)$, then $|\beta|=$

If the curves $\frac{x^2}{4}+y^2=1$ and $\frac{x^2}{a^2}+y^2=1$ for a suitable value of a cut on four concyclic points, the equation of the circle passing through these four points is

$P C P^{\prime}$ and $D C D^{\prime}$ are conjugate diameters of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\theta$ is the eccentric angle of $P$. Then, eccentric angle of the point where the circle $P P^{\prime} D$ again cuts the ellipse

Equation of a circle described on the Latusrectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ as a diameter with centre on the + ve $x$-axis is

Sum of the focal distance of any point on the ellipse $x^2 / a^2+y^2 / b^2=1$ is equal to the length of the

If $\mathrm{\theta }$ and $\phi$ are eccentric angle of the ends of a pair of conjugate diameters of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, then $\mathrm{\theta } - \phi$ is equal to

If the normal at point $P$ on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ meets the axes in $R$ and $S$ respectively, then $PR:RS$ is equal to

If the normal at any point ${P}$ on an ellipse of eccentricity '$e$' meets the major axis and minor axis at $T$ and $G$ respectively, then $PT : PG$ is equal to

If the eccentricity of the two ellipses $\frac{x^2}{169}+\frac{y^2}{25}=1$ and $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are equal, then the value of $a / b$ is