1
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1 Let $$y=x+2,4y=3x+6$$ and $$3y=4x+1$$ be three tangent lines to the circle $$(x-h)^2+(y-k)^2=r^2$$. Then $$h+k$$ is equal to :

A
6
B
5 (1 + $$\sqrt2$$)
C
5
D
5$$\sqrt2$$
2
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Let the tangents at the points $$A(4,-11)$$ and $$B(8,-5)$$ on the circle $$x^{2}+y^{2}-3 x+10 y-15=0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to

A
$$\frac{2\sqrt{13}}{3}$$
B
$$\frac{3\sqrt{3}}{4}$$
C
$$\sqrt{13}$$
D
$$2\sqrt{13}$$
3
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
$${x^2} + {y^2} + 3x + 3y + 4 = 0$$
4
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
$${\pi \over 6}$$
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