1
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1 If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x+y=a$$, is 16, then $$|a|$$ is equal to :

A
$$2 \sqrt{2}$$
B
$$2 \sqrt{3}$$
C
$$4 \sqrt{2}$$
D
4
2
JEE Main 2022 (Online) 27th July Morning Shift
+4
-1 Let $$P(a, b)$$ be a point on the parabola $$y^{2}=8 x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^{2}+y^{2}-10 x-14 y+65=0$$. Let $$A$$ be the product of all possible values of $$a$$ and $$B$$ be the product of all possible values of $$b$$. Then the value of $$A+B$$ is equal to

A
0
B
25
C
40
D
65
3
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1 Let $$\mathrm{P}$$ and $$\mathrm{Q}$$ be any points on the curves $$(x-1)^{2}+(y+1)^{2}=1$$ and $$y=x^{2}$$, respectively. The distance between $$P$$ and $$Q$$ is minimum for some value of the abscissa of $$P$$ in the interval

A
$$\left(0, \frac{1}{4}\right)$$
B
$$\left(\frac{1}{2}, \frac{3}{4}\right)$$
C
$$\left(\frac{1}{4}, \frac{1}{2}\right)$$
D
$$\left(\frac{3}{4}, 1\right)$$
4
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1 The acute angle between the pair of tangents drawn to the ellipse $$2 x^{2}+3 y^{2}=5$$ from the point $$(1,3)$$ is

A
$$\tan ^{-1}\left(\frac{16}{7 \sqrt{5}}\right)$$
B
$$\tan ^{-1}\left(\frac{24}{7 \sqrt{5}}\right)$$
C
$$\tan ^{-1}\left(\frac{32}{7 \sqrt{5}}\right)$$
D
$$\tan ^{-1}\left(\frac{3+8 \sqrt{5}}{35}\right)$$
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