1
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language

The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R}(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$\mathrm{P}$$ and $$\mathrm{Q}$$, then $$\frac{\beta^{2}}{\alpha}$$ is equal to _______________.

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2
JEE Main 2023 (Online) 6th April Morning Shift
Numerical
+4
-1
Out of Syllabus
Change Language

Let the tangent to the curve $$x^{2}+2 x-4 y+9=0$$ at the point $$\mathrm{P}(1,3)$$ on it meet the $$y$$-axis at $$\mathrm{A}$$. Let the line passing through $$\mathrm{P}$$ and parallel to the line $$x-3 y=6$$ meet the parabola $$y^{2}=4 x$$ at $$\mathrm{B}$$. If $$\mathrm{B}$$ lies on the line $$2 x-3 y=8$$, then $$(\mathrm{AB})^{2}$$ is equal to ___________.

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3
JEE Main 2023 (Online) 1st February Evening Shift
Numerical
+4
-1
Change Language

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 ____________.

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4
JEE Main 2023 (Online) 31st January Evening Shift
Numerical
+4
-1
Out of Syllabus
Change Language
Let $\mathrm{S}$ be the set of all $\mathrm{a} \in \mathrm{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(\mathrm{b}$, c), b, c $\in \mathbb{N}$, on the parabola $y^{2}=2 \mathrm{a} x$ and the lines $x=\mathrm{b}, y=0$ is $16 $ unit2, then $\sum\limits_{\mathrm{a} \in \mathrm{S}} \mathrm{a}$ is equal to :
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