Question

Let the coordinates of the points $A,B,C$ be $(1,8,4),(0,-11,4)$ and $(2,-3,1)$ respectively. What are the coordinates of the point $D$ which is the foot of the perpendicular from $A$ on $BC$?

• A. $(3,4,-2)$
• B. $(4,-2,5)$
• C. $(4,5,-2)$
• D. $(2,4,5)$

Answer 1

The correct option is C $(4,5,-2)$

given $A\equiv (1,8,4)$

$B\equiv (0,-11,4)$

$C\equiv (2,-3,1)$

Cartesian Equation of line passing through Pts $B$ and $C$ is given by

$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$

Now here substitute values, we get

$\to \frac{x-0}{2-0}=\frac{y+11}{-3+11}=\frac{z-4}{1-4}\Rightarrow \frac{x}{2}=\frac{y+11}{8}=\frac{z-4}{-3}$

Now the direction ratio of $AL$ is $(x-x_3),(y-y_3),(z-z_3)$

The coordinates of the points ${}^{\prime }{L}^{\prime }$ on line $BC$ is given by

$x=2\lambda ,y=8\lambda -11,z=-3\lambda +4$

$\therefore$ Directions ratios $\to (2\lambda -1),(8\lambda -19),(-3\lambda )$

Since $AL$ is perpendicular to Line $BC$

$\therefore a_1{a}_2+b_1{b}_2+c_1{c}_2=0$

$\to \left(2\right)(2\lambda -1)+8(8\lambda -19)-3(-3\lambda )=0$

$\to 4\lambda -2+64\lambda -152+9\lambda =0$

$\to 77\lambda =154\to \lambda =2$

Hence coordinates are

$\therefore \to (2\lambda ,8\lambda -11,-3\lambda +4)$

$\to 4,5,-2$.