1
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1

Let T and C respectively be the transverse and conjugate axes of the hyperbola $$16{x^2} - {y^2} + 64x + 4y + 44 = 0$$. Then the area of the region above the parabola $${x^2} = y + 4$$, below the transverse axis T and on the right of the conjugate axis C is :

A
$$4\sqrt 6 - {{28} \over 3}$$
B
$$4\sqrt 6 - {{44} \over 3}$$
C
$$4\sqrt 6 + {{28} \over 3}$$
D
$$4\sqrt 6 + {{44} \over 3}$$
2
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

Let the hyperbola $$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ pass through the point $$(2 \sqrt{2},-2 \sqrt{2})$$. A parabola is drawn whose focus is same as the focus of $$\mathrm{H}$$ with positive abscissa and the directrix of the parabola passes through the other focus of $$\mathrm{H}$$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $$\mathrm{H}$$, where e is the eccentricity of H, then which of the following points lies on the parabola?

A
$$(2 \sqrt{3}, 3 \sqrt{2})$$
B
$$\mathbf(3 \sqrt{3},-6 \sqrt{2})$$
C
$$(\sqrt{3},-\sqrt{6})$$
D
$$(3 \sqrt{6}, 6 \sqrt{2})$$
3
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
$$(2 \sqrt{5}, 3 \sqrt{6})$$
4
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)
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