Question

Find equations of altitudes and the coordinates of the orthocentre of the triangle whose sides are $3x-2y=6,3x+4y+12=0$ and $3x-8y+12=0$

Answer 1

$L_1:3x-2y-6=0$ ...(i)

$L_2:3x+4y+12=0$ ...(ii)

$L_1:3x-8y+12=0$ ...(iii)

Solving (i) & (ii)

$x=0,y=-3$

$\therefore B(0,-3)$

Solving (ii) & (iii), solving (i) & (iii)

$y=0,x=-4$ $y=3,x=4$

$\therefore C(-4,0)$ $A(4,3)$

Altitude ${}^{\prime }A{D}^{\prime }$ is $y-3=\frac{4}{3}(x-4)$

$\Rightarrow 4x-3y-7=0$ ...(iii)

Altitude ${}^{\prime }B{C}^{\prime }$ is

$y+3=\frac{-8}{3}(x-0)$

$\Rightarrow 8x+3y+9=0$ ...(iv)

Altitude ${}^{\prime }C{D}^{\prime }$ is

$y-0=\frac{-2}{3}(x+4)$

$\Rightarrow 3y=-2x-8$

$2x+3y+8=0$ ...(v)

Solving (iii) & (v)

$6x+1=0$

$x=\frac{-1}{6}$

$\therefore y=\frac{-23}{9}$

$\therefore$ orthocentre $(\frac{-1}{6},\frac{-23}{9})$