1
JEE Main 2023 (Online) 31st January Evening Shift
+4
-1
The set of all values of $a^{2}$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2 x^{2}+2 y^{2}-(1+a) x-(1-a) y=0$, is equal to :
A
$(0,4]$
B
$(4, \infty)$
C
$(2,12]$
D
$(8, \infty)$
2
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

Let a circle $$C_{1}$$ be obtained on rolling the circle $$x^{2}+y^{2}-4 x-6 y+11=0$$ upwards 4 units on the tangent $$\mathrm{T}$$ to it at the point $$(3,2)$$. Let $$C_{2}$$ be the image of $$C_{1}$$ in $$\mathrm{T}$$. Let $$A$$ and $$B$$ be the centers of circles $$C_{1}$$ and $$C_{2}$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the $$x$$-axis. Then the area of the trapezium AMNB is :

A
$$2\left( {2 + \sqrt 2 } \right)$$
B
$$4\left( {1 + \sqrt 2 } \right)$$
C
$$3 + 2\sqrt 2$$
D
$$2\left( {1 + \sqrt 2 } \right)$$
3
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$$\sqrt2$$
4
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
$$2\sqrt{13}$$
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