Question

The perpendicular distance of the point $P(1,2,3)$ from the straight line passing through the point $A(-1,4,7)$ and $B(2,8,7)$

• A. $\frac{\sqrt{149}}{25}$
• B. $\frac{149}{25}$
• C. $\frac{2\sqrt{149}}{25}$
• D. $\frac{2\sqrt{149}}{5}$

Answer 1

The correct option is D $\frac{2\sqrt{149}}{5}$

The drs of line joining points $A(-1,4,7)$ and $B(2,8,7)$ are $3,4,0$

The equation of line $AB$ is $\frac{x+1}{3}=\frac{y-4}{4}=\frac{z-7}{0}$

So the point on line $AB$ will look like $C(3t-1,4t+4,7)$

Assume that $CP$ is perpendicular to $AB$

So we get $(3t-2)3+(4t+2)4=0$

$\Rightarrow t=-\frac{2}{25}$

So the point $C$ is $(-\frac{31}{25}\frac{92}{25},7)$

The length of $CP$ is $\sqrt{{\left(\frac{56}{25}\right)}^2+{\left(\frac{42}{25}\right)}^2+7^2}=\frac{2\sqrt{149}}{5}$

Therefore the correct option is $D$