Question

A triangle has the lines $y=m_{1}x$ and $y=m_{2}x$ for two of its sides, where $m_1,m_2$ are the roots of the equation $x^2+ax-1=0$, then find the orthocentre of triangle.

Answer 1

$y=m_{1}x$ and $y=m_{2}x$ are two sides of triangle.

$x^2+ax-1=0$ has roots $m_1,m_2$

$\therefore m_1{m}_2=-1$

i.e., $y=m_{1}x\&y=m_{2}x$ lines are perpendicular to each other.

So, ortho center of right angle triangle lies at vertices of ${90}^{\circ }$ angle

So, orthocenter of given triangle is $(0,0)$