For a given line L: \( \displaystyle\frac{x}{2}=\displaystyle\frac{y+1}{3}=\displaystyle\frac{-z-3}{-5} \), which of the following is/are True ?
(A) The direction ratio of line L is (2, 3, -5)
(B) The perpendicular distance of the point (1, 1, 3) on the line L is \( \sqrt{3} \)
(C) The foot of perpendicular from the point (1, 1, 3) on line L is (2, 2, 1)
(D) The image of point (1, 1, 3) with respect to line L is (3, 3, 1)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (D) only
Since \( \frac{-z-3}{-5}=\frac{z+3}{5} \), the line in standard form is
\( \displaystyle \frac{x}{2}=\frac{y+1}{3}=\frac{z+3}{5}=t \)
with direction ratios (2, 3, 5). A general point on L is \( Q(2t,\ 3t-1,\ 5t-3) \).
For the foot of the perpendicular from P(1, 1, 3), PQ must be perpendicular to (2, 3, 5):
\( 2(2t-1)+3(3t-2)+5(5t-6)=0 \)
\( 38t-38=0 \)
\( t=1 \)
So the foot is Q(2, 2, 2).
\( PQ=\sqrt{1+1+1}=\sqrt{3} \)
The foot is the midpoint of P and its image, so the image is
\( (2\times 2-1,\ 2\times 2-1,\ 2\times 2-3)=(3,\ 3,\ 1) \)
So the direction ratios are (2, 3, 5), the foot is (2, 2, 2), the distance is \( \sqrt{3} \) and the image is (3, 3, 1).
Checking the statements
(A) The direction ratio of line L is (2, 3, -5) — Incorrect. In standard form the direction ratios are (2, 3, 5).
(B) The perpendicular distance of the point (1, 1, 3) on the line L is \( \sqrt{3} \) — Correct. The distance from (1, 1, 3) to the foot (2, 2, 2) is \( \sqrt{3} \).
(C) The foot of perpendicular from the point (1, 1, 3) on line L is (2, 2, 1) — Incorrect. The foot is (2, 2, 2).
(D) The image of point (1, 1, 3) with respect to line L is (3, 3, 1) — Correct. The foot (2, 2, 2) is the midpoint of (1, 1, 3) and (3, 3, 1).