Question

The coordinates of the foot of the perpendicular and the length of perpendicular drawn from \((0,2,3)\) to the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\) are respectively

• A. \((2,3,-1),\sqrt{31}\)
• B. \((2,3,1),\sqrt{21}\)
• C. \((2,3,-1),\sqrt{21}\)
• D. \((2,3,1),\sqrt{31}\)

Hint:

Take a general point on the line and use the condition that the joining vector is perpendicular to the line.

Answer 1

The Correct Option is C

Step 1: Check Option Feet
Test whether each given foot lies on the line: $(2,3,-1)$ gives $\dfrac{5}{5}=\dfrac{2}{2}=\dfrac{3}{3}=1$, so it lies on the line. $(2,3,1)$ does not, since $z$ would give $\dfrac{5}{3}$.

Step 2: Compute the Length
From $P(0,2,3)$ to $(2,3,-1)$: $\sqrt{4+1+16}=\sqrt{21}$.

Step 3: Confirm Perpendicularity
$(2,1,-4)\cdot(5,2,3)=10+2-12=0$, so it is perpendicular. Option (C).

Final Answer:
The foot (2, 3, -1) and length root 21 satisfy all conditions. \[ \boxed{\text{(C) }(2,3,-1),\sqrt{21}} \]