Ellipse practice · Mixed difficulty

If for the ellipse $\frac{{{x^2}}}{{{\alpha ^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, $y$-axis is the minor axis and the length of the latusrectum is one half of the length of its minor axis, then its eccentricity is

For the ellipse $25{x^2} + 9{y^2} - 150x - 90y + 225 = 0$, the eccentricity $e$ is equal to

The locus of the mid-points of focal chords of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,$ is

If $y=x+c$ is a normal to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, then $c^2$ is equal to

The minimum \& maximum distances of a point $(-1,6)$ from the ellipse $25 x^2+9 y^2+50 x+36 y-164=0$ are $l$ and $m$ respectively then $m^2-l^2$ equals

If $P$ is any point lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, whose foci are $S$ and $S^{\prime}$. Let $\angle P S S^{\prime}=\alpha$ and $\angle P S^{\prime} S=\beta$, then which of the following is/are true?

The number of values of c such that the line $y = 4x + c$ touches the curve $\frac{{{x^2}}}{4} + {y^2} = 1$ is

The number of maximum normals which can be drawn from a point to ellipse is

$3x^2 + 4y^2 – 6x + 8y + k = 0$ represents an ellipse with eccentricity $1/2,$

If the lengths of major and semi-minor axes of an ellipse are 4 and $\sqrt 3$ and their corresponding equations are $y - 5 = 0$ and $x + 3 = 0,$ then the equation of the ellipse is

The locus of the intersection of two normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which are perpendicular to each other, is

The equation of the ellipse with its centre at $(1,2)$, one focus at $(6,2)$ and passing through $(4,6)$, is

A line intersects the ellipse $\frac{x^2}{4 a^2}+\frac{y^2}{a^2}=1$ at $A$ and $B$ and the parabola $y^2=4 a(x+2 a)$ at $C$ and $D$. The line segment $A B$ subtends a right angle at the centre of the ellipse. Then, the locus of the point of intersection of tangents to the parabola at $C$ and $D$, is

Chord of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ touch $\frac{{{x^2}}}{{{\alpha ^2}}} + \frac{{{y^2}}}{{{\beta ^2}}} = 1,$ then the locus of their pole is

Which one of the following statements is not a false statement?

If $P C P^{\prime}$ and $D C D^{\prime}$ form a pair of conjugate diameters of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $R$ is any point on the circle $x^2+y^2=c^2$, then

The locus of the centroid of an equilateral triangle inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

The eccentricity of the conic $36{x^2} + 144{y^2} - 36x - 96y - 119 = 0$ is

Let $f(x)=x^2+9, g(x)=\frac{x}{x-9}$ and $a=fog(10), b=g o f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$, then $8 e^2+l^2$ is equal to.

If the area of the auxiliary circle of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1(a > b)$ is twice the area of the ellipse, then the eccentricity of the ellipse is

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\frac{x^2}{18}+\frac{y^2}{4}=1$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively. If the point $\left(e_1, e_2\right)$ lies on the ellipse $15 x^2+3 y^2=k$, then the value of $k$ is

Tangent is drawn to the ellipse $\frac{{{x^2}}}{{27}} + {y^2} = 1$ $\left( {3\sqrt 3 \cos \mathrm{\theta },\sin \mathrm{\theta }} \right)\left[ {\mathrm{where}\,\,\mathrm{\theta } \in \left( {0,\frac{\pi }{2}} \right)} \right]$. Then, the value of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is

The locus of the point of intersection of tangents to an ellipse at two points, sum of whose eccentric angles is constant is a/an

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

The equation of the ellipse whose axes are along the coordinate axes, vertices are $(\pm 5,0)$ are foci at $(\pm 4,0)$, is

If tangents drawn to the ellipse through point $(1,2 \sqrt{3})$ to the ellipse $x^2 / 9+y^2 / b^2=1$ are at right angle, then value of $b$ is

Length of major axis of ellipse $9{x^2} + 7{y^2} = 63$ is

If the chords of constant of tangents from two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ are at right angles, then $\frac{{{x_1}{x_2}}}{{{y_1}{y_2}}}$ is equal to

Equation of a tangent to the ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$, passing through the point where a directrix of the ellipse meets the + ve $x$-axis is

The eccentric angle of a point on the ellipse $9 x^2+4 y^2=36$ at a distance $\frac{\sqrt{31}}{2}$ units from the centre of the ellipse is '$\theta$', then the value of $|2 \cos \theta|$ is-