Question

Orthocentre of the triangle whose vertices are $(0,0),(2,-1)$ and $(1,3)$ is -

• A. $(-4,-1)$
• B. $(-\frac{4}{7},-\frac{1}{7})$
• C. $(\frac{4}{7},\frac{1}{7})$.
• D. $(4,1)$

Answer 1

The correct option is B $(-\frac{4}{7},-\frac{1}{7})$

Let the orthocenter of triangle be O (h,k)

Then The line AO$\perp$BC

product of slope=-1

$\therefore \frac{k}{h}\times \frac{3-(-1)}{1-2}=-1$

$\to \frac{k}{h}=\frac{1}{4}$

$\to h=4k-\left(1\right)$

Also line BO$\perp$AC

product of slope =-1

$\therefore \frac{-1-k}{2-h}\times \frac{3}{1}=-1$

$\to -3-3k=h-2$

$\to h+3k+1=0-\left(2\right)$

On solving eqs,(1) and (2) , we get, $h=\frac{-4}{7},k=\frac{-1}{7}$