1
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 ___________.

2
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 ___________.

3
JEE Main 2022 (Online) 24th June Evening Shift
Numerical
+4
-1

Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = ____________.

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

If two tangents drawn from a point ($$\alpha$$, $$\beta$$) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x are such that the slope of one tangent is four times the other, then the value of (10$$\alpha$$ + 5)2 + (16$$\beta$$2 + 50)2 equals ___________.

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