Question

The equations of the sides AB, BC and CA of ∆ ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0 respectively. The equation of the altitude through B is
(a) x − 3y + 1 = 0
(b) x − 3y + 4 = 0
(c) 3x − y + 2 = 0
(d) none of these

Answer 1

(b) x$-$3y = 4

The equation of the sides AB, BC and CA of ∆ABC are y − x = 2, x + 2y = 1 and 3x + y + 5 = 0, respectively.

Solving the equations of AB and BC, i.e. y − x = 2 and x + 2y = 1, we get:

x = − 1, y = 1

So, the coordinates of B are (−1, 1).

The altitude through B is perpendicular to AC.

$\therefore \mathrm{Slope}\mathrm{of}AC=-3$

$\mathrm{Thus},\mathrm{slope}\mathrm{of}\mathrm{the}\mathrm{altitude}\mathrm{through}B\mathrm{is}\frac{1}{3}.$

Equation of the required altitude is given below:

$$\begin{gathered} y-1=\frac{1}{3}\left(x+1\right) \\ \Rightarrow x-3y+4=0 \end{gathered}$$