Circle or vieta's?

Geometry Level 4

Let $\left (m_k,\frac{1}{m_k}\right)$ be four distinct points on a circle where $k=1,2,3,4$ , $m_k>0$ . Find $\displaystyle \prod_{k=1}^{4} m_k$ .

1 solution

Aditya Narayan Sharma
Mar 18, 2016

Let the equation be of the circle be $x^2+y^2+2gx+2fy+c=0$

Since $(m_k,\frac{1}{m_k})$ lies on it , it satisfies the equation.

Rearranging the equation we get, $m^4_k + 2gm^3_k+cm^2_k+2fm_k+1=0$

Clearly it's roots are $m_1,m_2,m_2,m_4$

So $\prod_{k=1}^{4} m_k=m_1m_2m_3m_4=1$ [By vieta's]