1
WB JEE 2023
+1
-0.25

If a hyperbola passes through the point P($$\sqrt2$$, $$\sqrt3$$) and has foci at ($$\pm$$ 2, 0), then the tangent to this hyperbola at P is

A
$$y = x\sqrt 6 - \sqrt 3$$
B
$$y = x\sqrt 3 - \sqrt 6$$
C
$$y = x\sqrt 6 + \sqrt 3$$
D
$$y = x\sqrt 3 + \sqrt 6$$
2
WB JEE 2023
+1
-0.25

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
$$a\{ 5\sqrt 2 + \ln (5 + \sqrt 2 )\}$$
3
WB JEE 2023
+1
-0.25

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 2022
+1
-0.25

Let $$P(3\sec \theta ,2\tan \theta )$$ and $$Q(3\sec \phi ,2\tan \phi )$$ be two points on $${{{x^2}} \over 9} - {{{y^2}} \over 4} = 1$$ such that $$\theta + \phi = {\pi \over 2},0 < \theta ,\phi < {\pi \over 2}$$. Then the ordinate of the point of intersection of the normals at P and Q is

A
$${{13} \over 2}$$
B
$$- {{13} \over 2}$$
C
$${5 \over 2}$$
D
$$- {5 \over 2}$$
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