Exam

JEE Main

Year

2012

Date

12 May

Mode

Online

Subject

Mathematics

Chapter

Straight Lines

JEE Main 2012 · 12 May · Online · Mathematics · Straight Lines

If two vertices of a triangle are (5,-1) and (-2,3) and its...

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

Step-by-step 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)$and (2) we geta=-4, b=-7$ Hence, third vertex is $(-4,-7)$.