An ellipse is inscribed in a circle and a point is inside a circle is choosen at random. If the

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

normal

An ellipse is inscribed in a circle and a point is inside a circle is choosen at random. If the probability that this point lies outside the ellipse is $\frac {2}{3}$ then eccentricity of ellipse is $\frac{{a\sqrt b }}{c}$ . Where $gcd( a, c) = 1$ and $b$ is square free integer ($b$ is not divisible by square of any integer except $1$ ) then $a · b · c$ is

A

$11$

B

$12$

C

$16$

D

$18$

Solution

Radius of circle $=$ semi-major axis

$p=\frac{\pi\left(a^{2}-a b\right)}{\pi a^{2}}=1-\frac{b}{a}=\frac{2}{3}$

$\Rightarrow \frac{b}{a}=\frac{1}{3}$

$\Rightarrow 1-\frac{b^{2}}{a^{2}}=\frac{8}{9}$

$\Rightarrow \mathrm{e}=\frac{2 \sqrt{2}}{3}$

So, $a \cdot b \cdot c=2 \cdot 2 \cdot 3=12$

Standard 11

Mathematics

Share

Similar Questions

The the circle passing through the foci of the $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1$ and having centre at $(0,3) $ is

medium

(JEE MAIN-2013)

A circle has the same centre as an ellipse and passes through the foci $F_1 \& F_2$ of the ellipse, such that the two curves intersect in $4$ points. Let $'P'$ be any one of their point of intersection. If the major axis of the ellipse is $17 $ and the area of the triangle $PF_1F_2$ is $30$, then the distance between the foci is :

normal

Eccentricity of the ellipse $4{x^2} + {y^2} – 8x + 2y + 1 = 0$ is

medium

The angle between the pair of tangents drawn to the ellipse $3{x^2} + 2{y^2} = 5$ from the point $(1, 2)$, is

hard

Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $\mathrm{A}$ and $\mathrm{B}$.

$1.$ The coordinates of $\mathrm{A}$ and $\mathrm{B}$ are

$(A)$ $(3,0)$ and $(0,2)$

$(B)$ $\left(-\frac{8}{5}, \frac{2 \sqrt{161}}{15}\right)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$

$(C)$ $\left(-\frac{8}{5}, \frac{2 \sqrt{161}}{15}\right)$ and $(0,2)$

$(D)$ $(3,0)$ and $\left(-\frac{9}{5}, \frac{8}{5}\right)$

$2.$ The orthocentre of the triangle $\mathrm{PAB}$ is

$(A)$ $\left(5, \frac{8}{7}\right)$ $(B)$ $\left(\frac{7}{5}, \frac{25}{8}\right)$

$(C)$ $\left(\frac{11}{5}, \frac{8}{5}\right)$ $(D)$ $\left(\frac{8}{25}, \frac{7}{5}\right)$

$3.$ The equation of the locus of the point whose distances from the point $\mathrm{P}$ and the line $\mathrm{AB}$ are equal, is

$(A)$ $9 x^2+y^2-6 x y-54 x-62 y+241=0$

$(B)$ $x^2+9 y^2+6 x y-54 x+62 y-241=0$

$(C)$ $9 x^2+9 y^2-6 x y-54 x-62 y-241=0$

$(D)$ $x^2+y^2-2 x y+27 x+31 y-120=0$

Give the answer question $1,2$ and $3.$

normal

(IIT-2010)