1
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1
Out of Syllabus

Let O be the origin and OP and OQ be the tangents to the circle $$x^2+y^2-6x+4y+8=0$$ at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point $$\left( {\alpha ,{1 \over 2}} \right)$$, then a value of $$\alpha$$ is :

A
1
B
$$-\frac{1}{2}$$
C
$$\frac{5}{2}$$
D
$$\frac{3}{2}$$
2
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1
Out of Syllabus

If the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the circle $$x^{2}+y^{2}-2 x+y=5$$ meet at the point $$R\left(\frac{9}{4}, 2\right)$$, then the area of the triangle $$\mathrm{PQR}$$ is :

A
$$\frac{13}{8}$$
B
$$\frac{5}{8}$$
C
$$\frac{5}{4}$$
D
$$\frac{13}{4}$$
3
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
$(8, \infty)$
4
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1
Out of Syllabus

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
$$2\left( {1 + \sqrt 2 } \right)$$
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