1
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
2
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1 Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of $${\pi \over 4}$$ with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is

A
8 only
B
2 only
C
$${1 \over 4}$$ only
D
any a > 0
3
JEE Main 2022 (Online) 29th June Morning Shift
+4
-1 Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of $${\pi \over 2}$$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $${a^2} > {b^2}$$. If e is the eccentricity of the ellipse E, then the value of $${1 \over {{e^2}}}$$ is equal to:

A
$$1 + \sqrt 2$$
B
$$3 + 2\sqrt 2$$
C
$$1 + 2\sqrt 3$$
D
$$4 + 5\sqrt 3$$
4
JEE Main 2022 (Online) 28th June Evening Shift
+4
-1 Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. Let e' and l' respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $${e^2} = {{11} \over {14}}l$$ and $${\left( {e'} \right)^2} = {{11} \over 8}l'$$, then the value of $$77a + 44b$$ is equal to :

A
100
B
110
C
120
D
130
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