1
WB JEE 2023
MCQ (Single Correct Answer)
+1
-0.25
Change Language

The average length of all vertical chords of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1,a \le x \le 2a$$, is :

A
$$b\{ 2\sqrt 3 - \ln (2 + \sqrt 3 )\} $$
B
$$b\{ 3\sqrt 2 + \ln (3 + \sqrt 2 )\} $$
C
$$a\{ 2\sqrt 5 - \ln (2 + \sqrt 5 )\} $$
D
$$a\{ 5\sqrt 2 + \ln (5 + \sqrt 2 )\} $$
2
WB JEE 2023
MCQ (Single Correct Answer)
+1
-0.25
Change Language

The tangent at point $$(a\cos \theta ,b\sin \theta ),0 < \theta < {\pi \over 2}$$, to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ meets the x-axis at T and y-axis at T$$_1$$. Then the value of $$\mathop {\min }\limits_{0 < \theta < {\pi \over 2}} (OT)(O{T_1})$$ is

A
ab
B
2ab
C
0
D
1
3
WB JEE 2023
MCQ (Single Correct Answer)
+1
-0.25
Change Language

Let $$A(2\sec \theta ,3\tan \theta )$$ and $$B(2\sec \phi ,3\tan \phi )$$ where $$\theta + \phi = {\pi \over 2}$$ be two points on the hyperbola $${{{x^2}} \over 4} - {{{y^2}} \over 9} = 1$$. If ($$\alpha,\beta$$) is the point of intersection of normals to the hyperbola at A and B, then $$\beta$$ is equal to

A
$${{12} \over 3}$$
B
$${{13} \over 3}$$
C
$$ - {{12} \over 3}$$
D
$$ - {{13} \over 3}$$
4
WB JEE 2023
MCQ (Single Correct Answer)
+1
-0.25
Change Language

If the lines joining the focii of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ where $$a > b$$, and an extremity of its minor axis is inclined at an angle 60$$^\circ$$, then the eccentricity of the ellipse is

A
$${{\sqrt 3 } \over 2}$$
B
$${1 \over 2}$$
C
$${{\sqrt 7 } \over 3}$$
D
$${1 \over {\sqrt 3 }}$$
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