If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and its orthocentre is at $(0,0)$, then the third…

If two vertices of a triangle are $(5,-1)$ and $(-2,3)$ and its orthocentre is at $(0,0)$, then the third vertex is
  1. $(4,-7)$
  2. $(-4,-7)$
  3. $(-4,7)$
  4. $(4,7)$

Solution


Let the third vertex of $\triangle A B C$ be $(a, b)$. Orthocentre $=H(0,0)$ Let $A(5,-1)$ and $B(-2,3)$ be other two vertices of $\triangle A B C$. Now, (Slope of $A H) \times($ Slope of $B C)=-1$ $ \begin{aligned} & \Rightarrow\left(\frac{-1-0}{5-0}\right)\left(\frac{b-3}{a+2}\right)=-1 \\ & \Rightarrow b-3=5(a+2) \end{aligned} $ Similarly, $($ Slope of $B H) \times($ Slope of $A C)=-1$ $ \begin{aligned} & \Rightarrow \quad-\left(\frac{3}{2}\right) \times\left(\frac{b+1}{a-5}\right)=-1 \\ & \Rightarrow \quad 3 b+3=2 a-10 \\ & \Rightarrow 3 b-2 a+13=0 \end{aligned} $ On solving equation (1) and (2) we get $a=-4, b=-7$ Hence, third vertex is $(-4,-7)$

Asked in: JEE Main 2012 (12 May Online)

Practice more Straight Lines questions on Aicharya