1
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
$$\frac{32}{9}$$
2
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
$$\frac{\sqrt{5}}{3}$$
3
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
$\frac{\sqrt{1741}}{5}$
4
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1
Out of Syllabus

Let the tangent and normal at the point $$(3 \sqrt{3}, 1)$$ on the ellipse $$\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$$ meet the $$y$$-axis at the points $$A$$ and $$B$$ respectively. Let the circle $$C$$ be drawn taking $$A B$$ as a diameter and the line $$x=2 \sqrt{5}$$ intersect $$C$$ at the points $$P$$ and $$Q$$. If the tangents at the points $$P$$ and $$Q$$ on the circle intersect at the point $$(\alpha, \beta)$$, then $$\alpha^{2}-\beta^{2}$$ is equal to :

A
61
B
$$\frac{304}{5}$$
C
60
D
$$\frac{314}{5}$$
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