Choose the correct answer. The equation of a circle with origin a… - Vidyadip

MCQ

Choose the correct answer. The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is:
[Hint: Centroid of the triangle coincides with the centre of the circle and the radius of the circle is $\frac{2}{3}$ of the length of the mediam]

A
$x^2+y^2=9 a^2$
B
$x^2+y^2=16 a^2$
✓
$x^2+y^2=4 a^2$
D
$x^2+y^2=a^2$

✓

Answer

Correct option: C.

$x^2+y^2=4 a^2$

$x^2+y^2=4 a^2$

Solution:
Let ABC be an equilateral triangle in which mediam AD = 3a
Centre of the circle is same as the centroid of the triangle i.e., $(0, 0)$

AG : GD = 2 : 1
So, $\text{AG}=\frac{2}{3}\text{AD}=\frac{2}{3}\times3\text{a}=2\text{a}$
$\therefore$ The equation of the circle is,
$(x - 0)^2 + (y - 0)^2= (2a)^2$
$\Rightarrow x^2 + y^2 = 4a^2$

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