Let mathrm{A}(α, 0) and mathrm{B}(0, β) be points on line 5 x+7 y=50 . Let point P divide line seg

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

Let $\mathrm{A}(\alpha, 0)$ and $\mathrm{B}(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide the line segment $A B$ internally in the ratio $7: 3$. Let $3 x-$ $25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $\mathrm{x}$-axis passes through $\mathrm{P}$, then the length of the latus rectum of $\mathrm{E}$ is equal to

A

$\frac{25}{3}$

B

$\frac{32}{9}$

C

$\frac{25}{9}$

D

$\frac{32}{5}$

(JEE MAIN-2024)

Solution

$\left.\begin{array}{l}\mathrm{A}=(10,0) \\ \mathrm{B}=\left(0, \frac{50}{7}\right)\end{array}\right\} \mathrm{P}=(3,5)$

$\mathrm{ae}=3$
$\frac{\mathrm{a}}{\mathrm{e}}=\frac{25}{3}$
$\mathrm{a}=5$
$\mathrm{~b}=4$
Length of $L R=\frac{2 b^2}{a}=\frac{32}{5}$

Standard 11

Mathematics

Share

Similar Questions

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

hard

(IIT-2003)

Let the ellipse $E : x ^2+9 y ^2=9$ intersect the positive $x$ – and $y$-axes at the points $A$ and $B$ respectively Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle which vertices $A, P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m – n$ is equal to

hard

(JEE MAIN-2023)

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)

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

Find the equation for the ellipse that satisfies the given conditions: Centre at $(0,\,0),$ major axis on the $y-$ axis and passes through the points $(3,\,2)$ and $(1,\,6)$

medium