1
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
$$\frac{347}{8}$$
2
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
$$\frac{32}{9}$$
3
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

A
$$\frac{1}{\sqrt{3}}$$
B
$$\frac{2}{\sqrt{5}}$$
C
$$\frac{\sqrt{3}}{2}$$
D
$$\frac{\sqrt{5}}{3}$$
4
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1, \frac{2}{5}\right)$, is equal to :
A
$\frac{\sqrt{1691}}{5}$
B
$\frac{\sqrt{2009}}{5}$
C
$\frac{\sqrt{1541}}{5}$
D
$\frac{\sqrt{1741}}{5}$
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