Question

In a triangle $\Delta ABC$, the sum of the lengths of two sides is represented by $p$ and the product of the lengths of the same two sides is $q$. Let $c$ be the third side. If $p^2=c^2+2q$, then the area of the circumcircle of $\Delta ABC$ is

• A. $\frac{\pi c^2}{2}$
• B. $2\pi c^2$
• C. $\frac{\pi c^2}{4}$
• D. $\pi c^2$

Answer 1

The correct option is C $\frac{\pi c^2}{4}$

Let the two sides of the triangle are $a$and $b$
Then $p=a+b$ and $q=ab$

Now, $p^2=c^2+2q$
$\Rightarrow a^2+b^2+2ab=c^2+2ab$
$\Rightarrow a^2+b^2=c^2$
$\Rightarrow \Delta ABC$ is right-angled triangle, where $c$ is the length of hypotenuse.
So, radius of the circumcircle is $\frac{c}{2}$
$\therefore$ Area of the circumcircle is $\frac{\pi c^2}{4}$.