Question

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
(a) x² + y² = 9a²
(b) x² + y² = 16a²
(c) x² + y² = 4a²
(d) x² + y² = a²

Answer 1

Since origin the center for circle
⇒ x² + y² = r² is the required equation and since above circle passes through the vertices of equilateral triangle with medium 3a. The centroid of an equilateral triangle is the center of its circumcircle and the radius of the circle is the distance of any vertex from centroid.
$\begin{array}{rcl}\mathrm{i}.\mathrm{e}.\mathrm{radius}\mathrm{of}\mathrm{circle}& =& \mathrm{distance}\mathrm{of}\mathrm{centroid}\mathrm{from}\mathrm{any}\mathrm{vertex}\\ & =& \frac{2}{3}\left(\mathrm{median}\right)\\ & =& \frac{2}{3}\left(3a\right)\left(\mathrm{given}\mathrm{medium}=3a\right)\end{array}$
i.e. radius of circle = 2a
∴ equation of circle is x² + y² = (2a)²
i.e. x² + y² = 4a²
Hence, the correct answer is option C.