1
JEE Main 2023 (Online) 13th April Evening Shift
+4
-1 Let the centre of a circle C be $$(\alpha, \beta)$$ and its radius $$r < 8$$. Let $$3 x+4 y=24$$ and $$3 x-4 y=32$$ be two tangents and $$4 x+3 y=1$$ be a normal to C. Then $$(\alpha-\beta+r)$$ is equal to :

A
7
B
9
C
5
D
6
2
JEE Main 2023 (Online) 10th April Evening Shift
+4
-1 Let A be the point $$(1,2)$$ and B be any point on the curve $$x^{2}+y^{2}=16$$. If the centre of the locus of the point P, which divides the line segment $$\mathrm{AB}$$ in the ratio $$3: 2$$ is the point C$$(\alpha, \beta)$$, then the length of the line segment $$\mathrm{AC}$$ is :

A
$$\frac{3 \sqrt{5}}{5}$$
B
$$\frac{6 \sqrt{5}}{5}$$
C
$$\frac{2 \sqrt{5}}{5}$$
D
$$\frac{4 \sqrt{5}}{5}$$
3
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1 A line segment AB of length $$\lambda$$ moves such that the points A and B remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :

A
$${2 \over 3}\lambda$$
B
$${3 \over 5}\lambda$$
C
$${{\sqrt {19} } \over 7}\lambda$$
D
$${{\sqrt {19} } \over 5}\lambda$$
4
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1 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
$$\frac{3}{2}$$
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