Question

If a $\Delta ABC$ has vertices $A(-1,7),B(-7,1)$ and $C(5,-5)$, then its orthocentre has coordinates:

• A. $(3,-3)$
• B. $\left(-\frac{3}{5},\frac{3}{5}\right)$
• C. $(-3,3)$
• D. $\left(\frac{3}{5},-\frac{3}{5}\right)$

Answer 1

Answer: (C)

Let orthocentre is $H\left(x_0,y_0\right)$

$\Rightarrow \left(\frac{y_0-7}{x_0+1}\right)\left(\frac{1+5}{-7-5}\right)=-1$
$\Rightarrow 2x_0-y_0+9=0$ .....(1)
and $m_{BH}\cdot m_{AC}=-1$
$\Rightarrow \left(\frac{y_0-1}{x_0+7}\right)\left(\frac{7-(-5)}{-1-5}\right)=-1$
$\Rightarrow x_0-2y_0+9-0$.....(2)
Solving equation (1) and (2) we get
$\left(x_0,y_0\right)\equiv (-3,3)$