Ellipse

Assertion (A) The length of the latus rectum of an ellipse is 4 . The focus and its corresponding directrix are respectively $(1,-2)$ and $3 x+4 y-15=0$. Then, its eccentricity is $\frac{1}{2}$.
Reason $(\mathrm{R})$ Length of the perpendic...

If the normal at the point $P\left(\frac{\pi}{4}\right)$ on the ellipse $x^2+4 y^2-4=0$ meets the ellipse again at $Q(\alpha, \beta)$, then $\alpha=$

If $P(\alpha, \beta)$ is a point on the curve $9 x^2+4 y^2=144$ in the first quadrant and the minimum area of the triangle formed by the tangent of the curve at $P$ with the coordinate axis is $S$, then

If a tangent having slope $\frac{1}{3}$ to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$ is a normal to the circle $(x+1)^2+(y+1)^2=1$, then $a^2$ lies in the interval

The area (in sq. units) of the triangle formed by the tangent and normal to the ellipse $9 x^2+4 y^2=72$ at the point $(2,3)$ with the $X$-axis is

If the tangents drawn from a point $P$ to the ellipse $4 x^2+9 y^2-16 x+54 y+61=0$ are perpendicular, then the locus of $P$ is

The equation of the normal drawn at the point $(\sqrt{2}+1,-1)$ to the ellipse $x^2+2 y^2-2 x+8 y+5=0$ is

The angle between the tangents drawn from a point $(-3,2)$ to the ellipse $4 x^2+9 y^2-36=0$ is

Let $A_1$ be the area of the given ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Let $A_2$ be the area of the region bounded by the curve which is the locus of mid-point of the line segment joining the focus of the ellipse and a point $P$ on ...

The square of the slope of a common tangent drawn to the circle $4 x^2+4 y^2=25$ and the ellipse $4 x^2+9 y^2=36$ is

The equation of a chord $A B$ of an ellipse $2 x^2+y^2=1$ is $x-y+1=0$. If $O$ is the origin, then $\sqrt{A O B}=$

If the tangents are drawn to the ellipse $x^2+2 y^2=2$, then the locus of the mid-points of the intercepts made by the tangents between the coordinate axes is

Let $T_1$ be the tangent drawn at a point $P(\sqrt{2}, \sqrt{3})$ on the ellipse $\frac{x^2}{4}+\frac{y^2}{6}=1$. If ( $\alpha, \beta$ ) is the point where, $T_1$ intersects another tangent $T_2$ to the ellipse perpendicularly, then $\alpha...

The length of the latusrectum of $16 x^2+25 y^2=400$ is

If $A_1, A_2, A_3$ are the areas of ellipse $x^2+4 y^2-4=0$ its director circle and auxiliary circle respectively, then $A_2+A_3-A_1=$

The product of perpendiculars from the two foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ on the tangent at any point on the ellipse is

Let F and $F^1$ be the foci of the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1(b<2)$ and $B$ is one end of the minor axis. If the area of the triangle $\mathrm{FBF}^1$ is $\sqrt{3}$ sq units, then the eccentricity of the ellipse is

If the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ having $(1,1)$ as its middle point is $x+\alpha y=\beta$, then

If a tangent of slope 2 to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touches the circle $x^2+y^2=4$, then maximum value of $a b$ is

If $4 x-3 y-5=0$ is a normal to the ellipse $3 x^2+8 y^2=k$, then the equation of the tangent drawn to this ellipse at the point $(-2, m)(m>0)$ is

Let $x^2+y^2=20$ be the director circle of an ellipse $E$ whose major axis is $X$-axis and minor axis is $Y$-axis. If the length of the latusrectum of $E$ is 2 . Then, the distance between its foci is

Let the length of the latusrectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be equal to the length of its semi-major axis. If the radius of its director circle is $\sqrt{3}$ and $e$ is its eccentricity, then the length of its latusr...

If the angle between the straight lines joining the foci and the ends of the minor axis of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is \(90^{\circ}\), then it eccentricity

A stick of length \(r\) units slides with its ends on coordinate axes. Then, the locus of the mid-point of the stick is a curve whose length is

The eccentric angle of a point on the ellipse \(x^2+3 y^2=6\) lying at a distance of 2 units from its centre is

The focal distances of the point \(\left(\frac{4}{\sqrt{5}}, \frac{3}{\sqrt{5}}\right)\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{9}=1\) are

If \(\tan \theta_1, \tan \theta_2=\frac{-a^2}{b^2}\), then the chord joining 2 points \(\theta_1\) and \(\theta_2\) one the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) will subtend a right angle at

A point moves so that the sum of its distances from \((a e, 0)\) and \((-a e, 0)\) is \(2 a\), then the equation to its locus, where \(b^2=a^2\left(1-e^2\right)\) is

If a point \(P(x, y)\) moves along the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\) and if \(C\) is the center of the ellipse, then the sum of maximum and minimum values of \(C P\) is

In an ellipse, if the distance between the foci
is 6 units and the length of its minor axis is
8 units, then its eccentricity is