1
JEE Main 2022 (Online) 26th June Evening Shift
Numerical
+4
-1

Let a line L1 be tangent to the hyperbola $${{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1$$ and let L2 be the line passing through the origin and perpendicular to L1. If the locus of the point of intersection of L1 and L2 is $${({x^2} + {y^2})^2} = \alpha {x^2} + \beta {y^2}$$, then $$\alpha$$ + $$\beta$$ is equal to _____________.

2
JEE Main 2022 (Online) 26th June Morning Shift
Numerical
+4
-1

Let the common tangents to the curves $$4({x^2} + {y^2}) = 9$$ and $${y^2} = 4x$$ intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $${l \over {{e^2}}}$$ is equal to ______________.

3
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $${5 \over 4}$$. If the equation of the normal at the point $$\left( {{8 \over {\sqrt {5} }},{{12} \over {5}}} \right)$$ on the hyperbola is $$8\sqrt 5 x + \beta y = \lambda$$, then $$\lambda$$ $$-$$ $$\beta$$ is equal to ___________.

4
JEE Main 2022 (Online) 24th June Evening Shift
Numerical
+4
-1

Let the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$$ and the ellipse $$E:3{x^2} + 4{y^2} = 12$$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If $${e_H}$$ and $${e_E}$$ are the eccentricities of H and E respectively, then the value of $$12\left( {e_H^2 + e_E^2} \right)$$ is equal to ___________.

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