Ellipse practice · Mixed difficulty

$P$ is a variable point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with $A A^{\prime}$ as the major axis. Then, the maximum value of the area of the triangle APA', is

Ellipses which are drawn with the same two perpendicular lines as axes and with the sum of the reciprocals of squares of the lengths of their semi-major axis and semi-minor axis equal to a constant have only

The locus of a point whose distance from the point $(3,0)$ is $3 / 5$ times its distance from the line $x=p$ is an ellipse with centre at the origin. The value of $p$ is

The equation $\sqrt{(x-2)^2+y^2}+\sqrt{(x+2)^2+y^2}=4$ represents

If $e_1$ is the eccentricity of the conic $9 x^2+4 y^2=36$ and $e_2$ is the eccentricity of the conic $9 x^2-4 y^2$ $=36$, then -

In the standard form of an ellipse sum of the focal distances of a point 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

If $p$ and $p^{\prime}$ denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length $a$, respectively, on a tangent to the ellipsic and $r$ denotes the focal distance of the point, then

The distance between the foci of the ellipse $5{x^2} + 9{y^2} = 45,$ is

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 $d_1$ and $d_2$ be the lengths of perpendiculars drawn from foci S and $S^{\prime}$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to the tangent at any point $P$ on the ellipse. Then, $SP : SP' =$

The coordinates of the foci of the ellipse $4x^2 + 9y^2 = 1$ are

The equation $\frac{x^2}{2-r}+\frac{y^2}{r-5}+1=0$ represents an ellipse if

If $a{x^2} + b{y^2} + 2\mathrm{g}x + 2fy + c = 0$ represents an ellipse, then

If polar of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is always touching the parabola $y^2=4 a x$, then locus of the pole is

The locus of the point of intersection of mutually perpendicular tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

The values of $\lambda$ for which the line $y=x+\lambda$ touches the ellipse $9 x^2+16 y^2=144$, are

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 $\triangle P^{\prime} Q^{\prime} R^{\prime}=$

The equation of the chord of contact of the pair of tangents drawn to the ellipse $4 x^2+9 y^2=36$ from the point $(m, n)$, where $m \cdot n=m+n, m, n$ being non-zero positive integers is

If $y=m x+4$ is a tangent to the ellipse $\frac{x^2}{25}+\frac{y^2}{4}=1$, then $625 m^4+25 m^2-156$ is equal to

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 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 $\theta_1, \theta_2, \theta_3, \theta_4$ 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

Eccentricity of hyperbola $\frac{{{x^2}}}{k} + \frac{{{y^2}}}{{{k^2}}} = 1\,(k < 0)$ is

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

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1$ from any of its foci?

If the normals at $P\left(x_1, y_1\right), Q\left(x_2, y_2\right)$ and $R\left(x_3, y_3\right)$ to the ellipse are concurrent, then

If $C P$ and $CD$ are semi-conjugate diameters of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then $C P^2+C D^2=$

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