1
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1 Let a line L pass through the point of intersection of the lines $$b x+10 y-8=0$$ and $$2 x-3 y=0, \mathrm{~b} \in \mathbf{R}-\left\{\frac{4}{3}\right\}$$. If the line $$\mathrm{L}$$ also passes through the point $$(1,1)$$ and touches the circle $$17\left(x^{2}+y^{2}\right)=16$$, then the eccentricity of the ellipse $$\frac{x^{2}}{5}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$$ is :

A
$$\frac{2}{\sqrt{5}}$$
B
$$\sqrt{\frac{3}{5}}$$
C
$$\frac{1}{\sqrt{5}}$$
D
$$\sqrt{\frac{2}{5}}$$
2
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1
Out of Syllabus 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
$$\tan ^{-1}\left(\frac{3+8 \sqrt{5}}{35}\right)$$
3
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1 If the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ meets the line $$\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$$ on the $$x$$-axis and the line $$\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1$$ on the $$y$$-axis, then the eccentricity of the ellipse is :

A
$$\frac{5}{7}$$
B
$$\frac{2 \sqrt{6}}{7}$$
C
$$\frac{3}{7}$$
D
$$\frac{2 \sqrt{5}}{7}$$
4
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2} - {a^2}{y^2} = 5$$, where a is the minimum distance between the curves y = ex and y = logex. Then $${a^2} + {1 \over {{b^2}}}$$ is equal to :

A
$${3 \over 2}$$
B
$${5 \over 2}$$
C
3
D
5
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