1
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1 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 ___________.

2
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1 Two tangent lines $$l_{1}$$ and $$l_{2}$$ are drawn from the point $$(2,0)$$ to the parabola $$2 \mathrm{y}^{2}=-x$$. If the lines $$l_{1}$$ and $$l_{2}$$ are also tangent to the circle $$(x-5)^{2}+y^{2}=r$$, then 17r is equal to ___________.

3
JEE Main 2022 (Online) 28th July Morning Shift
Numerical
+4
-1 For the hyperbola $$\mathrm{H}: x^{2}-y^{2}=1$$ and the ellipse $$\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$$, a $$>\mathrm{b}>0$$, let the

(1) eccentricity of $$\mathrm{E}$$ be reciprocal of the eccentricity of $$\mathrm{H}$$, and

(2) the line $$y=\sqrt{\frac{5}{2}} x+\mathrm{K}$$ be a common tangent of $$\mathrm{E}$$ and $$\mathrm{H}$$.

Then $$4\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)$$ is equal to _____________.

4
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1 A common tangent $$\mathrm{T}$$ to the curves $$\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ and $$C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$$ does not pass through the fourth quadrant. If $$\mathrm{T}$$ touches $$\mathrm{C}_{1}$$ at $$\left(x_{1}, y_{1}\right)$$ and $$\mathrm{C}_{2}$$ at $$\left(x_{2}, y_{2}\right)$$, then $$\left|2 x_{1}+x_{2}\right|$$ is equal to ______________.

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