1
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
$$\frac{4 \sqrt{5}}{5}$$
2
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
$${{\sqrt {19} } \over 5}\lambda$$
3
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
$$\frac{3}{2}$$
4
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
$$\frac{13}{4}$$
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