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

If the line $$x-1=0$$ is a directrix of the hyperbola $$k x^{2}-y^{2}=6$$, then the hyperbola passes through the point :

A
$$(-2 \sqrt{5}, 6)$$
B
$$(-\sqrt{5}, 3)$$
C
$$(\sqrt{5},-2)$$
D
$$(2 \sqrt{5}, 3 \sqrt{6})$$
2
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1
Out of Syllabus

Let the tangent drawn to the parabola $$y^{2}=24 x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2 x+2 y=5$$. Then the normal to the hyperbola $$\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$$ at the point $$(\alpha+4, \beta+4)$$ does NOT pass through the point :

A
(25, 10)
B
(20, 12)
C
(30, 8)
D
(15, 13)
3
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1

Let the foci of the ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$$ and the hyperbola $$\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$$ coincide. Then the length of the latus rectum of the hyperbola is :

A
$$\frac{32}{9}$$
B
$$\frac{18}{5}$$
C
$$\frac{27}{4}$$
D
$$\frac{27}{10}$$
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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