Ellipse practice · Mixed difficulty

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

$P$ is any point on the ellipse $81{x^2} + 144{y^2} = 1944$, whose foci are $S$ and $S'$. Then, $SP + S'P$ equals

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 ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5 b+7$, then $b$ belongs to

The point $(4,-3)$ w.r.t. the ellipse $4 x^2+5 y^2=1$

For the ellipse $3 x^2+4 y^2=12$, the length of latus rectum is

The equation $2 x^2+3 y^2=30$ represents

If $S$ and $S^{\prime}$ are two foci of the ellipse $16 x^2+25 y^2=400$ and PSQ is a focal chord such that $S P=16$, then $S Q=$

Area of quadrilateral formed by joining the focii of the ellipse, $\frac{x^2}{16}+\frac{y^2}{12}=1$ and $\frac{x^2}{12}+\frac{y^2}{16}=1$ is (in sq. units)

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

If $y=x$ and $3 y+2 x=0$ are the equations of a pair of conjugate diameters of an ellipse, then the eccentricity of the ellipse is

In triangle $\mathrm{ABC}$ given the base $\mathrm{BC}$ of the triangle and if $\angle C-\angle{B}=K$, a constant. Then the locus of the vertex $\mathrm{A}$ is

Equation of the ellipse whose axes are the axes of co-ordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$, is

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

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

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:

If the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ coincide with the foci of the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{81}=\frac{1}{25}$, then $b^{2}$ 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 line $12x\cos \theta \, + \,5y\sin \theta \, = \,60$ is tangent to which of the following curves?

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

If $A$ and $B$ are two fixed points and $P$ is a variable point such that $PA + PB = 4,$ the locus of $P$ is

The eccentricity of the ellipse $9{x^2} + 5{y^2} - 30y = 0$ is

The line $y = x + 1$ meets the ellipse $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1$ at two points P and Q. If r is the radius of the circle with PQ as diameter then ${(3r)^2}$ is equal to:

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

If $\alpha$ and $\beta$ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse, is

The sum of the focal distances from any point on the ellipse $9{x^2} + 16{y^2} = 144$ is

If the focal distance of an end of the minor axis of an ellipse (referred to its axes as the axes of $x$ and $y$ respectively) is $k$ and the distance between its foci is $2h$, then its equation is