Question

Find the equation of line passing through the points A (0, 6, −9) and B (−3, −6, 3). If D is the foot of perpendicular drawn from a point C (7, 4, −1) on the line AB, then find the coordinates of the point D and the equation of line CD.

Answer 1

Equation of line AB passing through the points A(0, 6,$-$9) and B($-$3, $-$6, 3) is

$$\begin{gathered} \frac{x-0}{-3-0}=\frac{y-6}{-6-6}=\frac{z+9}{3+9} \\ =\frac{x}{1}=\frac{y-6}{4}=\frac{z+9}{-4} \end{gathered}$$

Here, D is the foot of the perpendicular drawn from C (7, 4, $-$1) on AB.

The coordinates of a general point on AB are given by

$$\begin{gathered} \frac{x}{1}=\frac{y-6}{4}=\frac{z+9}{-4}=\lambda \\ \Rightarrow x=\lambda \\ y=4\lambda +6 \\ z=-4\lambda -9 \end{gathered}$$

Let the coordinates of D be $\left(\lambda ,4\lambda +6,-4\lambda -9\right)$.
The direction ratios of CD are proportional to $\lambda -7,4\lambda +6-4,-4\lambda -9+1,\mathrm{i}.\mathrm{e}.\lambda -7,4\lambda +2,-4\lambda -8$.

The direction ratios of AB are proportional to 1, 4, $-$4, but CD is perpendicular to AB.

Substituting $\lambda =-1$ in $\left(\lambda ,4\lambda +6,-4\lambda -9\right)$, we get the coordinates of D as $\left(-1,2,-5\right)$.

Equation of CD is
$$\begin{gathered} \frac{x-7}{-1-7}=\frac{y-4}{2-4}=\frac{z+1}{-5+1} \\ =\frac{x-7}{4}=\frac{y-4}{1}=\frac{z+1}{2} \end{gathered}$$