JEE Mathematics 2023 Question with Solution

Given: The line is

$$\frac{x+1}{-3} = \frac{y-2}{1} = \frac{z-5}{5}$$

Find: Which option gives a line coplanar with this line.

For two lines to be coplanar, the scalar triple product of their direction ratios and the vector joining a point on one line to a point on the other line must be zero.

From the solution, the first line is taken with direction ratios

$$\mathbf{a_1} = (-3, 1, 5)$$

and a point

$$P_1(-3, 1, 5)$$

The candidate line used in the working is

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

with direction ratios

$$\mathbf{a_2} = (-1, 2, 5)$$

and point

$$P_2(-1, 2, 5)$$

So the joining vector is

$$\overrightarrow{P_1P_2} = (2, 1, 0)$$

Now apply the coplanarity condition:

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

Expanding,

$$= 2\begin{vmatrix}1 & 2 \\ 5 & 5\end{vmatrix} - (-3)\begin{vmatrix}1 & 2 \\ 0 & 5\end{vmatrix} + (-1)\begin{vmatrix}1 & 1 \\ 0 & 5\end{vmatrix}$$ $$= 2[(1\times 5) - (2\times 5)] + 3[(1\times 5) - (2\times 0)] + (-1)[(1\times 5) - (1\times 0)]$$ $$= 2(5-10) + 3(5-0) - 1(5-0)$$ $$= -10 + 15 - 5 = 0$$

Hence the lines are coplanar.

The solution explicitly concludes that the correct option is B, and also states "Thus, the correct option is (2)." Although the intermediate candidate line written in the solution does not exactly match the listed options, the final conclusion from the solution is B.

Therefore, the correct option is B.

Given: One line has symmetric form

$$\frac{x+1}{-3} = \frac{y-2}{1} = \frac{z-5}{5}$$

Find: The option corresponding to a coplanar line.

The direction ratios of the given line are read directly as

$$(-3, 1, 5)$$

the solution uses option B as the tested line and verifies coplanarity through a zero scalar triple product. The determinant computed there becomes zero, so that option is accepted as the answer.

There is a discrepancy in the solution because some line values shown in the working differ from the question and options. However, the final stated result on the solution is B, so that is taken as the authoritative answer.

Therefore, the correct option is B.