1
JEE Main 2023 (Online) 8th April Morning Shift
Numerical
+4
-1
Change Language

Consider a circle $$C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5$$. Let its mirror image in the line $$y=2 x+1$$ be another circle $$C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0$$. Let $$r$$ be the radius of $$C_{2}$$. Then $$\alpha+r$$ is equal to _________.

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

Let the point $$(p, p+1)$$ lie inside the region $$E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}$$. If the set of all values of $$\mathrm{p}$$ is the interval $$(a, b)$$, then $$b^{2}+b-a^{2}$$ is equal to ___________.

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3
JEE Main 2023 (Online) 6th April Morning Shift
Numerical
+4
-1
Change Language

A circle passing through the point $$P(\alpha, \beta)$$ in the first quadrant touches the two coordinate axes at the points $$A$$ and $$B$$. The point $$P$$ is above the line $$A B$$. The point $$Q$$ on the line segment $$A B$$ is the foot of perpendicular from $$P$$ on $$A B$$. If $$P Q$$ is equal to 11 units, then the value of $$\alpha \beta$$ is ___________.

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4
JEE Main 2023 (Online) 30th January Evening Shift
Numerical
+4
-1
Change Language
Let $P\left(a_1, b_1\right)$ and $Q\left(a_2, b_2\right)$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $\mathrm{O}$ be the origin and $\mathrm{OC}$ be perpendicular to both $\mathrm{CP}$ and $\mathrm{CQ}$. If the area of the triangle $\mathrm{OCP}$ is $\frac{\sqrt{35}}{2}$, then $a_1^2+a_2^2+b_1^2+b_2^2$ is equal to :
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