Question

The orthocentre of the triangle with vertices $(-2,6),(-2,4)and(1,3)$ is

• A. $(-3,3)$
• B. $(-1,1/3)$
• C. $(1,3)$
• D. none of these

Answer 1

The correct option is A $(-3,3)$

$\underset{A}{(-2,6)},\underset{B}{(-2,4)},\underset{C}{(1,3)}$

slope of $BC=\frac{3-4}{1-(-2)}$

$m_1=\frac{-1}{3}$

equation of line $\perp BC$ passing through A.

$m_1{m}_2=-1m_2=3$

$A(-2,6)$

$y=3x+C$

$6=3(-2)+C$

$C=12$

$y=3x+12$ ... (i)

slope of $AC=\frac{3-6}{1-(-2)}$

$=\frac{-3}{3}=-1$

equation of line $\perp AC$ passing through B

$m_1{m}_2=-1m_1=2$

$B(-2,4)$

$y=x+c$

$4=-2+c$

$c=6$

$y=x+6$ ... (ii)

Intersection of (i) and (ii)

$y=3x+12$

$-(y=x+6)$

____________

$0=2x+6$

____________

$x=-3\&y=-3+6$

$x=-3\&y=3$

orthocentre $(-3,3)$.