Match List-I with List-II
| List-I | List-II | ||
| Equation of line | Direction ratios | ||
| (A) | \( 2x=3y=-z \) | (I) | 2, -3, -2 |
| (B) | \( \displaystyle\frac{x-2}{2}=\frac{2y-5}{-3},\ z=2 \) | (II) | 3, 2, -6 |
| (C) | \( \displaystyle\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z-3}{2} \) | (III) | 4, -3, 0 |
| (D) | \( \displaystyle\frac{x+1}{2}=\frac{2-y}{3}=\frac{z+3}{-2} \) | (IV) | 2, -3, 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (II), (B) - (III), (C) - (IV), (D) - (I)
The direction ratios are the denominators once each line is in the form \( \frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c} \), with x, y, z each having coefficient +1 in the numerator.
(A) \( 2x=3y=-z \)
This matches (II). Dividing by 6:
\( \displaystyle \frac{x}{3}=\frac{y}{2}=\frac{z}{-6} \)
so the direction ratios are 3, 2, −6.
(B) \( \displaystyle\frac{x-2}{2}=\frac{2y-5}{-3},\ z=2 \)
This matches (III). Dividing the numerator and denominator of the y-part by 2:
\( \displaystyle \frac{x-2}{2}=\frac{y-\frac{5}{2}}{-\frac{3}{2}} \)
z is constant, so its direction ratio is 0. The ratios 2, \( -\frac{3}{2} \), 0 are the same as 4, −3, 0.
(C) \( \displaystyle\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z-3}{2} \)
This matches (IV). It is already in standard form: 2, −3, 2.
(D) \( \displaystyle\frac{x+1}{2}=\frac{2-y}{3}=\frac{z+3}{-2} \)
This matches (I). Since \( \frac{2-y}{3}=\frac{y-2}{-3} \):
\( \displaystyle \frac{x+1}{2}=\frac{y-2}{-3}=\frac{z+3}{-2} \)
so the direction ratios are 2, −3, −2.
So the matching is (A) - (II), (B) - (III), (C) - (IV), (D) - (I).