Question

The area of an equilateral triangle inscribed in the circle x² + y² − 6x − 8y − 25 = 0 is
(a) $\frac{225\sqrt{3}}{6}$

(b) 25π

(c) 50π − 100

(d) none of these

Answer 1

(a) $\frac{225\sqrt{3}}{6}$



Let ABC be the required equilateral triangle.
The equation of the circle is x² + y² − 6x − 8y − 25 = 0.
Therefore, coordinates of the centre O is $\left(3,4\right)$.

Radius of the circle = OA = OB = OC = $\sqrt{9+16+25}=5\sqrt{2}$

In $∆$BOD, we have:

$$\begin{gathered} \sin 60°=\frac{DB}{BO} \\ \Rightarrow DB=\frac{\sqrt{3}}{2}\left(5\sqrt{2}\right) \\ \Rightarrow BC=2BD=\sqrt{3}\left(5\sqrt{2}\right)=5\sqrt{6} \end{gathered}$$

Now, area of $△ABC$ = $\frac{\sqrt{3}}{4}B{C}^2=\frac{\sqrt{3}}{4}{\left(5\sqrt{6}\right)}^2=\frac{\sqrt{3}\left(150\right)}{4}=\frac{\sqrt{3}\left(75\right)}{2}=\frac{\sqrt{3}\left(225\right)}{6}$ square units