1
JEE Main 2024 (Online) 1st February Evening Shift
+4
-1
Let $\mathrm{P}$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $\mathrm{P}$ and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point $\mathrm{Q}$ such that $\mathrm{P}$ and $\mathrm{Q}$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $P Q$ such that $P R: R Q=4: 3$ as $P$ moves on the ellipse, is :
A
$\frac{13}{21}$
B
$\frac{\sqrt{139}}{23}$
C
$\frac{\sqrt{13}}{7}$
D
$\frac{11}{19}$
2
JEE Main 2024 (Online) 1st February Morning Shift
+4
-1
Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a}>\mathrm{b}$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is :
A
3
B
$${7 \over 2}$$
C
$${3 \over 2}$$
D
$${5 \over 2}$$
3
JEE Main 2024 (Online) 31st January Evening Shift
+4
-1

Let $$P$$ be a parabola with vertex $$(2,3)$$ and directrix $$2 x+y=6$$. Let an ellipse $$E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$$, of eccentricity $$\frac{1}{\sqrt{2}}$$ pass through the focus of the parabola $$P$$. Then, the square of the length of the latus rectum of $$E$$, is

A
$$\frac{512}{25}$$
B
$$\frac{656}{25}$$
C
$$\frac{385}{8}$$
D
$$\frac{347}{8}$$
4
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$A(\alpha, 0)$$ and $$B(0, \beta)$$ 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 $$x$$-axis passes through $$P$$, then the length of the latus rectum of $$E$$ is equal to,

A
$$\frac{25}{3}$$
B
$$\frac{25}{9}$$
C
$$\frac{32}{5}$$
D
$$\frac{32}{9}$$
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