Question

The co-ordinates of the foot of the perpendicular drawn from the point A$(1,0,3)$ to the join of the point B$(4,7,1)$ and C$(3,5,3)$ are?

• A. $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$
• B. $(5,7,17)$
• C. $(\frac{5}{3},-\frac{7}{3}\frac{17}{3})$
• D. $(-\frac{5}{3},\frac{7}{3},-\frac{17}{3})$

Answer 1

The correct option is A $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$

Equation of the line $AB$

$\frac{x-4}{3-4}=\frac{y-7}{5-7}=\frac{z-1}{3-1}$

$\frac{x-4}{-1}=\frac{y-7}{-2}=\frac{z-1}{2}$

Consider,

$\frac{x-4}{-1}=\frac{y-7}{-2}=\frac{z-1}{2}=r$

$x=-r+4,y=-2r+7,z=2r+1$

General point on the line $AB$ is $(-r+4,-2r+7,2r+1)$

Direction ratios of perpendicular are

$-r+4-1,-2r+7-0,2r+1-3$

That is

$-r+3,-2r+7,2r-2$

since perpendicular is on the line $AB$

Therefore,

$-1(-r+3)-2(-2r+7)+2(2r-2)=0$

then,

$r=\frac{7}{3}$

Therefore,

coordinates of the foot of perpendicular are

$\left(\frac{5}{3},\frac{7}{3},\frac{17}{3}\right)$