1
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

Let $$e_1$$ be the eccentricity of the hyperbola $$\frac{x^2}{16}-\frac{y^2}{9}=1$$ and $$e_2$$ be the eccentricity of the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a} > \mathrm{b}$$, which passes through the foci of the hyperbola. If $$\mathrm{e}_1 \mathrm{e}_2=1$$, then the length of the chord of the ellipse parallel to the $$x$$-axis and passing through $$(0,2)$$ is :

A
$$\frac{8 \sqrt{5}}{3}$$
B
$$3 \sqrt{5}$$
C
$$4 \sqrt{5}$$
D
$$\frac{10 \sqrt{5}}{3}$$
2
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1

Let R be a rectangle given by the lines $$x=0, x=2, y=0$$ and $$y=5$$. Let A$$(\alpha,0)$$ and B$$(0,\beta),\alpha\in[0,2]$$ and $$\beta\in[0,5]$$, be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. Then, the mid-point of AB lies on a :

A
hyperbola
B
straight line
C
parabola
D
circle
3
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1
Out of Syllabus

Let $$\mathrm{P}\left(x_{0}, y_{0}\right)$$ be the point on the hyperbola $$3 x^{2}-4 y^{2}=36$$, which is nearest to the line $$3 x+2 y=1$$. Then $$\sqrt{2}\left(y_{0}-x_{0}\right)$$ is equal to :

A
3
B
$$-$$9
C
$$-$$3
D
9
4
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
Let $\mathrm{H}$ be the hyperbola, whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :
A
$\frac{5}{2}$
B
3
C
2
D
$\frac{3}{2}$
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