Ellipse practice · Medium

Let $P$ be a point on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,$ of eccentricity $e$. If $A,A'$ are the vertices and $S,S'$ are the foci of the ellipse, then Area $\Delta PSS':\mathrm{Area}\,\,\Delta APA' =$

If $\theta_1, \theta_2, \theta_3, \theta_4$ are eccentric angles of 4 concyclic points on Ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ Then $\cos \left(\frac{\theta_1+\theta_2+\theta_3+\theta_4}{4}\right)$ cannot be

The locus of the poles of the chords of the ellipse subtending a right angle at its centre is

If the normals to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the ends of the chords $l_1 x+m_1 y=1$ and $l_2 x+m_2 y=1$ are concurrent, then

Tangent are drawn from any point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to the circle $x^2+y^2=r^2$, then the chord of contact is a tangent to the ellipse, is

Three points $A,B,C$ are taken on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with eccentric angles $\mathrm{\theta },\mathrm{\theta } + \mathrm{\alpha }\,\mathrm{and }\,\mathrm{\theta } + 2\mathrm{\alpha }$, then

Equation of the latusrectum of the ellipse $9{x^2} + 4{y^2} - 18x - 8y - 23 = 0\;$ are

The equation of ellipse, whose distance between the foci is equal to 8 and the distance between the directrices is 18 , is

An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms. Are

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ be a given ellipse, length of whose latusrectum is 10 . If its eccentricity is the maximum value of the function $\phi(t)=\frac{5}{12}+t-t^2$, then $a^2+b^2$ is equal to

The product of the perpendiculars from the foci of the ellipse $\frac{x^2}{144}+\frac{y^2}{100}=1$ on any tangent is:

If a triangle is inscribed in an ellipse and two of its sides are parallel to given straight lines, then envelope of the third side is

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

The tangent at the point $P\left(\frac{4}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right)$ to the ellipse $9 x^2+16 y^2=144$ meets the axis of $x$ at $A$ and the axis of $y$ at $B$. If $C$ is the centre of the ellipse, then area of the $\Delta A B C$ is (in sq. units)

The locus of points whose polars with respect to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ are at a distance $d$ from the centre of the ellipse, is

If roots of quadratic equation $a{x^2} + 2bx + c = 0$ are not real, then $a{x^2} + 2bxy + c{y^2} + dx + ey + f = 0$ represents a/an

The eccentricity of the ellipse $25 x^2+16 y^2=400$ is

Let $P_i$ and $P_i^{\prime}$ be the feet of the perpendiculars drawn from the foci $S$ and $S^{\prime}$ on a tangent $T_i$ to an ellipse whose length of semi-major axis is 20 . If $\sum\limits_{{i=1}}^{10}\left(S P_i\right)\left(S^{\prime} P_i^{\prime}\right)=2560$, then the value of eccentricity is

Chords of an ellipse are drawn through the positive end of the minor axis. Then, their mid-point lies on

If the tangents on the ellipse $4{x^2} + {y^2} = 8$ at the points (1, 2) and $(a,\,b)$ are perpendicular to each other, then ${a^2}$ is equal to

The length of the major axis of the ellipse $(5 x-10)^2+(5 y+15)^2=\frac{(3 x-4 y+7)^2}{4}$ is

If the point of intersection of the ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=1$ are at the extremities of the conjugate diameters of the former, then

The minimum distance of the centre of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ from the chord of contact of mutually perpendicular tangents of the ellipse is

If the chord of contact of tangents drawn from a point $P$ to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ subtends a right-angle at its centre, then $P$ lies on

A man running round a race course notes that the sum of the distances of two flag posts from him is always 10m and the distance between the flag posts is 8m. The area of the path he encloses (in square metre) is

Tangent at a point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is drawn which cuts the coordinates axes at $A$ and $B$. The minimum area of the triangle $O A B$ is ( $O$ being origin)

The foci of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$ and the hyperbola $\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}}$ coincide. Then, the value of ${b^2}$ is

If the normal at the end of latusrectum of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ passes through $\left( {0,\; - b} \right)$, then ${e^4} + {e^2}$ (where $e$ is eccentricity) equals

Let $\left( {\mathrm{\alpha },\mathrm{\beta }} \right)$ be a point from which two perpendicular tangents can be drawn to the ellipse $4{x^2} + 5{y^2} = 20.$ If $F = 4\mathrm{\alpha } + 3\mathrm{\beta }$, then

$P$ is a point on the circle ${x^2} + {y^2} = {c^2}.$ The locus of the mid-points of chords of contact of $P$ with respect to $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,$ is