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

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 ____________.

2
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1

Let C be the largest circle centred at (2, 0) and inscribed in the ellipse $${{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1$$. If (1, $$\alpha$$) lies on C, then 10 $$\alpha^2$$ is equal to ____________

3
JEE Main 2023 (Online) 24th January Morning Shift
Numerical
+4
-1
Out of Syllabus

Let a tangent to the curve $$9{x^2} + 16{y^2} = 144$$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is ________

4
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1
Out of Syllabus

Let the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the ellipse $$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$ meet at the point $$R(\sqrt{2}, 2 \sqrt{2}-2)$$. If $$\mathrm{S}$$ is the focus of the ellipse on its negative major axis, then $$\mathrm{SP}^{2}+\mathrm{SQ}^{2}$$ is equal to ___________.

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