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

Two vertices of a triangle are $(5,-1)$ and $(-2,3)$. If the origin is the orthocentre of this triangle, then the coordinates of the third vertex of that triangle are
  1. (4, 7)
  2. $\left(-2, \frac{-7}{2}\right)$
  3. $(-4,-7)$
  4. $(-2,3)$

Solution

Slope of $B C \times$ slope of $A D=-1$
$\Rightarrow \quad \frac{3+1}{-2-5} \times \frac{k}{h}=-1 \Rightarrow \frac{k}{h} \times \frac{4}{-7}=-1$
Slope of $A B \times$ slope of $C F=-1$ $ \begin{aligned} & \Rightarrow M_{A B} \times M_{C F}=-1\left\{\begin{array}{l} \because M_{C F} \\ \frac{3-0}{-2-0}=\frac{-3}{2} \end{array}\right\} \\ & \Rightarrow M_{A B} \times \frac{-3}{2}=-1 \Rightarrow M_{A B}=\frac{2}{3} \end{aligned} $ Equation of $A B$ $ \begin{array}{rlrl} & & y+1 & =\frac{2}{3}(x-5) \\ \Rightarrow & 3 y+3 & =2 x-10 \\ \Rightarrow & 2 x-3 y & =13 \end{array} $ $A(h, k)$ lie on line $A B$
From Eqs. (i) and (ii), we get $ h=-4, k=-7 $ Hence, third vertex of the triangle $=(-4,-7)$

Asked in: AP EAMCET 2019 (21 Apr Shift 1)

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