Question

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If $x^2-c^2=y$,where c is the length of the third side of the triangle, then the circumradius of the triangle is :

• A. $\frac{y}{13}$
• B. $\frac{c}{\sqrt{3}}$
• C. $\frac{c}{3}$
• D. $\frac{3}{2}y$

Answer 1

The correct option is B $\frac{c}{\sqrt{3}}$

Given $a+b=x$ and $ab=y$

If $x^2-c^2=y\Rightarrow (a+b{)}^2-c^2=ab$

$\Rightarrow a^2+b^2-c^2=-ab$

$\Rightarrow \frac{a^2+b^2-c^2}{2ab}=-\frac{1}{2}$

$\Rightarrow cosC=-\frac{1}{2}$

$\Rightarrow \angle C=\frac{2\pi }{3}$

$R=\frac{C}{2sinC}=\frac{C}{\sqrt{3}}$