Ellipse practice · Easy

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

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

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

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

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

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

In the ellipse, the minor axis is 8 and eccentricity is $\sqrt{5} / 3$. Then major axis is

If $\alpha$ and $\beta$ are eccentric angles of the ends of $a$ focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$ is equal to

The normal at the point $(3,2 \sqrt{3})$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ meets the major axis of the ellipse at $(\alpha, 0)$, the value of $\alpha$ is:

In an ellipse length of minor axis is 8 and eccentricity is $\frac{{\sqrt 5 }}{3}$. The length of major axis is

Locus of the point which divides double ordinate of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ in the ratio $1:2$ internally, is

The vertices of the ellipse $9 x^2+4 y^2-18 x-27=0$, are

A focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$, then length of semi major axis is

All ellipse has its centre at $\left( {1, - 1} \right)$ and semi-major axis $= 8$ and it passes through the point $\left( {1,3} \right)$. The equation of the ellipse is

The equation of the ellipse whose foci are $\left( { \pm \,2,0} \right)$ and eccentricity $\frac{1}{2}$ is