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

The points of intersection of the line $$ax + by = 0,(a \ne b)$$ and the circle $${x^2} + {y^2} - 2x = 0$$ are $$A(\alpha ,0)$$ and $$B(1,\beta )$$. The image of the circle with AB as a diameter in the line $$x + y + 2 = 0$$ is :

A
$${x^2} + {y^2} + 5x + 5y + 12 = 0$$
B
$${x^2} + {y^2} + 3x + 5y + 8 = 0$$
C
$${x^2} + {y^2} - 5x - 5y + 12 = 0$$
D
$${x^2} + {y^2} + 3x + 3y + 4 = 0$$
2
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1

The locus of the mid points of the chords of the circle $${C_1}:{(x - 4)^2} + {(y - 5)^2} = 4$$ which subtend an angle $${\theta _i}$$ at the centre of the circle $$C_1$$, is a circle of radius $$r_i$$. If $${\theta _1} = {\pi \over 3},{\theta _3} = {{2\pi } \over 3}$$ and $$r_1^2 = r_2^2 + r_3^2$$, then $${\theta _2}$$ is equal to :

A
$${\pi \over 2}$$
B
$${\pi \over 4}$$
C
$${{3\pi } \over 4}$$
D
$${\pi \over 6}$$
3
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1
Out of Syllabus

Let the tangents at two points $$\mathrm{A}$$ and $$\mathrm{B}$$ on the circle $$x^{2}+\mathrm{y}^{2}-4 x+3=0$$ meet at origin $$\mathrm{O}(0,0)$$. Then the area of the triangle $$\mathrm{OAB}$$ is :

A
$$\frac{3 \sqrt{3}}{2}$$
B
$$\frac{3 \sqrt{3}}{4}$$
C
$$\frac{3}{2 \sqrt{3}}$$
D
$$\frac{3}{4 \sqrt{3}}$$
4
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

For $$\mathrm{t} \in(0,2 \pi)$$, if $$\mathrm{ABC}$$ is an equilateral triangle with vertices $$\mathrm{A}(\sin t,-\cos \mathrm{t}), \mathrm{B}(\operatorname{cost}, \sin t)$$ and $$C(a, b)$$ such that its orthocentre lies on a circle with centre $$\left(1, \frac{1}{3}\right)$$, then $$\left(a^{2}-b^{2}\right)$$ is equal to :

A
$$\frac{8}{3}$$
B
8
C
$$\frac{77}{9}$$
D
$$\frac{80}{9}$$
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