JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

\text{The line that is coplanar to the line } \frac{x + 3}{-3} = \frac{y - 1}{1} = \frac{z - 5}{5} \text{ is:}

Choose the correct answer:

Explanation

1. Given Line ( $L_1$ ):

\frac{x + 3}{-3} = \frac{y - 1}{1} = \frac{z - 5}{5}

Point $P_1 = (-3, 1, 5)$
Direction Ratio $\vec{d_1} = (-3, 1, 5)$

2. Option (1) ko check karte hain:

Line: $\frac{x + 1}{-1} = \frac{y - 2}{2} = \frac{z - 5}{5}$

Point $P_2 = (-1, 2, 5)$
Direction Ratio $\vec{d_2} = (-1, 2, 5)$
$P_2 - P_1 = (-1 - (-3), 2 - 1, 5 - 5) = (2, 1, 0)$

Determinant check:

\Delta = \begin{vmatrix} 2 & 1 & 0 \\ -3 & 1 & 5 \\ -1 & 2 & 5 \end{vmatrix}

= 2(5 - 10) - 1(-15 - (-5)) + 0

= 2(-5) - 1(-10) = -10 + 10 = 0

Kyuki determinant $0$ aa gaya hai, iska matlab Option (1) wali line coplanar hai.

Final Answer

Sahi option (1) hai.

Correct Option: (1)

\frac{x + 1}{-1} = \frac{y - 2}{2} = \frac{z - 5}{5}