JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Circle

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 AB in the ratio

3 : 2

is the point

C (α, β)

, then the length of the line segment AC is

(A) $\frac{6 5}{5}$
(B) $\frac{4 5}{5}$
(C) $\frac{2 5}{5}$
(D) $\frac{3 5}{5}$ ✓ Correct

Answer: (D)

Solution

Take

B = (4 cos θ, 4 sin θ)

. Since

P

divides

A B

with

A (1, 2)

in the ratio

3 : 2

P = (\frac{12 cos θ + 2}{5}, \frac{12 sin θ + 4}{5})

. Writing

P = (h, k)

gives

12 cos θ = 5 h - 2

and

12 sin θ = 5 k - 4

, so

(5 h - 2)^{2} + (5 k - 4)^{2} = 144

, a circle centred at

(\frac{2}{5}, \frac{4}{5})

. Thus

C = (\frac{2}{5}, \frac{4}{5})

and

A C = (\frac{3}{5})^{2} + (\frac{6}{5})^{2} = \frac{45}{25} = \frac{3 5}{5}

. The length of

A C

\frac{3 5}{5}

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Circle →

More JEE Main questions on Circle

Let PQ and MN be two straight lines touching the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ at the points A and B respectively. Let O be the centre of the circle and $∠ AOB = π /3$ . Then the locus of the point of intersection of the lines PQ and MN is:
JEE Main 2026View solution →
If P is a point on the circle $x^{2} + y^{2} = 4$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ______.
JEE Main 2026View solution →
Let the set of all values of r, for which the circles $(x + 1)^{2} + (y + 4)^{2} = r^{2}$ and $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval ( $α$ , $β$ ). Then $α β$ is equal to
JEE Main 2026View solution →
Let a circle of radius 4 pass through the origin O , the points $A (- 3 a, 0)$ and $B (0, - 2 b)$ , where a and b are real parameters and $ab \neq = 0$ . Then the locus of the centroid of $△ OAB$ is a cir …
JEE Main 2026View solution →
Let $y = x$ be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord …
JEE Main 2026View solution →
Let the circle $x^{2} + y^{2} = 4$ intersect $x$ -axis at the points $A (a, 0), a > 0$ and $B (b, 0)$ . Let $P (2 cos α, 2 sin α), 0 < α < \frac{π}{2}$ and $Q (2 cos β, 2 sin β)$ be two points …
JEE Main 2026View solution →