1
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1

If the $$x$$-intercept of a focal chord of the parabola $$y^{2}=8x+4y+4$$ is 3, then the length of this chord is equal to ____________.

2
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1

The line $$x=8$$ is the directrix of the ellipse $$\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ with the corresponding focus $$(2,0)$$. If the tangent to $$\mathrm{E}$$ at the point $$\mathrm{P}$$ in the first quadrant passes through the point $$(0,4\sqrt3)$$ and intersects the $$x$$-axis at $$\mathrm{Q}$$, then $$(3\mathrm{PQ})^{2}$$ is equal to ____________.

3
JEE Main 2023 (Online) 29th January Evening Shift
Numerical
+4
-1

A triangle is formed by the tangents at the point (2, 2) on the curves $$y^2=2x$$ and $$x^2+y^2=4x$$, and the line $$x+y+2=0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to ___________.

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

The vertices of a hyperbola H are ($$\pm$$ 6, 0) and its eccentricity is $${{\sqrt 5 } \over 2}$$. Let N be the normal to H at a point in the first quadrant and parallel to the line $$\sqrt 2 x + y = 2\sqrt 2$$. If d is the length of the line segment of N between H and the y-axis then d$$^2$$ is equal to _____________.

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