JEE Main 2026 — Jan 24, Shift 1 — Mathematics solution

JEE Main2026Jan 24, Shift 1Mathematics

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 $△ O A B$ is a circle of radius

A
$\frac{5}{3}$
B
$\frac{7}{3}$
C
$\frac{8}{3}$
D
$\frac{11}{3}$

Solution

Since $A$ lies on the x-axis and $B$ lies on the y-axis, $∠ A O B = 9 0^{\circ}$ , so $A B$ is a diameter of the circle. Hence $∣ A B ∣ = 2 \times 4 = 8$ and

$∣ A B ∣^{2} = 3 a^{2} + 2 b^{2} = 64 \dots (*)$

Let the centroid be $G (h, k)$ :

$h = \frac{0 + ( - 3 a ) + 0}{3} = - \frac{3 a}{3}, k = \frac{0 + 0 + ( - 2 b )}{3} = - \frac{2 b}{3}$

So $a = - 3 h$ and $b = - \frac{3 k}{2}$ .

Substituting into $(*)$ :

$3 (3 h^{2}) + 2 \cdot \frac{9 k ^{2}}{2} = 64 \Rightarrow 9 h^{2} + 9 k^{2} = 64 \Rightarrow h^{2} + k^{2} = \frac{64}{9}$ .

The locus is a circle of radius $\frac{8}{3}$ .

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