question_answer

The equation of circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length\[3a\], is

A)  \[{{x}^{2}}+{{y}^{2}}=9{{a}^{2}}\]

B)  \[{{x}^{2}}+{{y}^{2}}=16{{a}^{2}}\]

C)  \[{{x}^{2}}+{{y}^{2}}=16{{a}^{2}}\]

D)  \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\]

Correct Answer: C

Solution :

Key Idea: Centroid divides the median in the ratio of\[2:1\]. Given, circle has centre at origin and passing through vertices of equilateral \[\Delta \] whose median is of length\[3a\]. \[\therefore \]Radius\[=3a\cdot \frac{2}{3}=2a\] \[\Rightarrow \]Equation of circle                 \[{{(x-0)}^{2}}+{{(y-0)}^{2}}={{(2a)}^{2}}\] \[\Rightarrow \]               \[{{x}^{2}}+{{y}^{2}}=4{{a}^{2}}\]