1
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

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
2
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
$\frac{3}{2}$
3
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

If the maximum distance of normal to the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is :

A
$$\frac{\sqrt{3}}{4}$$
B
$$\frac{1}{2}$$
C
$$\frac{1}{\sqrt{2}}$$
D
$$\frac{\sqrt{3}}{2}$$
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then

$$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$$ :

A
is an empty set
B
contains exactly one element
C
contains exactly two elements
D
is an infinite set
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