Let a circle of radius 4 pass through the origin O, the points and where and are real parameters and . Then the locus of the centroid of is a circle of radius [2026]

Question

Let a circle of radius 4 pass through the origin O, the points $A (- \sqrt{3} a, 0)$ and $B (0, - \sqrt{2} b),$ where $a$ and $b$ are real parameters and $a b \neq 0$ . Then the locus of the centroid of $∆ O A B$ is a circle of radius [2026]

Smart Achievers · Accepted Answer

(2)

$A B = 8$

$3 a^{2} + 2 b^{2} = 64$

$a$

$h = - \frac{\sqrt{3 a}}{3}, k = - \frac{\sqrt{2 b}}{3}$

$a = - \sqrt{3} h, b = - \frac{3}{\sqrt{2}} k$

$9 h^{2} + 9 k^{2} = 64$

$x^{2} + y^{2} = \frac{64}{9}$

$r = \frac{8}{3}$

Solution

Q.
Let a circle of radius 4 pass through the origin O, the points $A (- \sqrt{3} a, 0)$ and $B (0, - \sqrt{2} b),$ where $a$ and $b$ are real parameters and $a b \neq 0$ . Then the locus of the centroid of $∆ O A B$ is a circle of radius [2026]

1 $\frac{7}{3}$

2 $\frac{8}{3}$

3 $\frac{5}{3}$

4 $\frac{11}{3}$

Ans.
(2)

$A B = 8$

$3 a^{2} + 2 b^{2} = 64$

$Centroid G (h, k)$

$h = - \frac{\sqrt{3 a}}{3}, k = - \frac{\sqrt{2 b}}{3}$

$a = - \sqrt{3} h, b = - \frac{3}{\sqrt{2}} k$

$9 h^{2} + 9 k^{2} = 64$

$x^{2} + y^{2} = \frac{64}{9}$

$r = \frac{8}{3}$

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let a circle of radius 4 pass through the origin O, the points A(-3a,0) and B(0,-2b), where a and b are real parameters and ab≠0. Then the locus of the centroid of ∆OAB is a circle of radius [2026]

1 73

2 83

3 53

4 113

Ans. (2) AB=8 3a2+2b2=64 Centroid G(h,k) h=-3a3, k=-2b3 a=-3 h, b=-32 k 9h2+9k2=64 x2+y2=649 r=83