1
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1
Out of Syllabus

The distance of the point $$(6,-2\sqrt2)$$ from the common tangent $$\mathrm{y=mx+c,m > 0}$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :

A
$$\frac{1}{3}$$
B
5
C
$$\frac{14}{3}$$
D
5$$\sqrt3$$
2
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1
Out of Syllabus

The equations of the sides AB and AC of a triangle ABC are $$(\lambda+1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda)y+\lambda=0$$ respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola $$y^2=6x$$ in the first quadrant is :

A
4
B
2$$\sqrt2$$
C
2
D
$$\sqrt6$$
3
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1
Out of Syllabus

Let a tangent to the curve $$\mathrm{y^2=24x}$$ meet the curve $$xy = 2$$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

A
length of latus rectum 2
B
directrix 4x = $$-$$3
C
directrix 4x = 3
D
length of latus rectum $$\frac{3}{2}$$
4
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$\mathrm{H}: x^{2}-y^{2}=4$$. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is :

A
$$2 \sqrt{6}$$
B
$$2 \sqrt{14}$$
C
$$4 \sqrt{6}$$
D
$$4 \sqrt{14}$$
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