Two vertices of a triangle are $(5,-1)$, and $(-2,3)$. If the orthocentre of the triangle is the ori

Question

Two vertices of a triangle are $(5,-1)$, and $(-2,3)$. If the orthocentre of the triangle is the origin, find the coordinates of the third vertex.

Professor PC Thomas Classes · Answer

$H C \perp A B \Rightarrow\left(\frac{b}{a}ight)\left(\frac{4}{-7}ight)=-1$ $$\begin{equation*}\Rightarrow \quad 4 b=7 a 	ag{1} \end{equation*}$$ $A H \perp B C \Rightarrow\left(-\frac{1}{5}ight)\left(\frac{3-b}{-2-a}ight)=-1$ $$\begin{equation*}\Rightarrow \quad 5 a+13=b 	ag{2} \end{equation*}$$ From (1) \& (2), $a=-4, b=-7$\ $	herefore \quad$ Third vertex $(-4,-7)$.\ \