iCON Education - Ellipse AP-EAPCET Test

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 $\alpha=$

$\frac{-23}{17 \sqrt{2}}$

$\frac{7 \sqrt{2}}{17}$

$\frac{1}{\sqrt{2}}$

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 perpendicular drawn from focus of an ellipse to its corresponding directrix is $\frac{a\left(1-e^2\right)}{e}$.

Then, which one of the following is correct?

(A) and (R) are true and (R) is the correct explanation of (A)

(A) and (R) are true and (R) is not the correct explanation of (A)

(A) is true, (R) is false

(A) is false, (R) is true

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

$\left(\frac{\sqrt{2}}{\sqrt{5}}, 2\right)$

$\left(\frac{2}{5}, 4\right)$

$\left(1, \frac{10}{9}\right)$

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

$S=\sqrt{\alpha \beta}$

$S=2 \sqrt{\alpha \beta}$

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

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

$\sqrt{2} x-y=3+\sqrt{2}$

$2 x+y=2 \sqrt{2}+1$

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

$x^2+y^2-4 x+6 y+4=0$

$x^2+y^2-6 x+4 y+9=0$

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 given ellipse, then $A_1: A_2=$

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

$\tan ^{-1}\left(\frac{2}{3}\right)$

$\tan ^{-1}\left(\frac{3}{2}\right)$

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 $\sqrt{A O B}=$

$\tan ^{-1}\left(\frac{1}{2}\right)$

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

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

$\frac{x^2}{4}+\frac{y^2}{2}=1$

$\frac{x^2}{2}+\frac{y^2}{4}=1$

$\frac{1}{4 x^2}+\frac{1}{2 y^2}=1$

$\frac{1}{2 x^2}+\frac{1}{4 y^2}=1$

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 $\alpha^2+\beta^2$ is equal to

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

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

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 $A_2+A_3-A_1=$

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

$2 \alpha-1=3 \beta$

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

$\frac{\sqrt{3}}{2}$ or $\frac{1}{2}$

$\frac{1}{\sqrt{3}}$

$\frac{\sqrt{3}}{4}$ or $\frac{1}{4}$

$\frac{3}{4}$ or $\frac{1}{4}$

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

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

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

$\frac{10}{3}, \frac{2}{3}$

$\frac{13}{3}, \frac{5}{3}$

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

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

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

$\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$

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

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

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