Question

Find the equation of a line passing through $(1,2,3)$ having direction ratios $1,2,4$

• A. $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}=r$
• B. $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-4}{3}=r$
• C. $\frac{x-1}{1}=\frac{y-2}{1}=\frac{z-4}{3}=r$
• D. $\frac{x-1}{1}=\frac{y-2}{1}=\frac{z-3}{4}=r$

Answer 1

The correct option is D $\frac{x-1}{1}=\frac{y-2}{1}=\frac{z-3}{4}=r$

We want to find the equation of a line and we are given the direction ratios and a point on the line. To find the line, let’s consider a general point $P(x,y,z)$ on the same line.
Now the direction ratios of the line segment connecting P and Q should be proportional to $1,2,4$
Direction ratio of line joining P and Q =$x-1,y-2,z-3$
From question, direction ratios of the line = 1, 2, 4
Since these are proportional,
$\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{4}=r$
Since any point on this line satisfies this condition, this is the equation of the line. This way of writing the equation of a line is called symmetrical form of a line.