JEE Mathematics 2025 Question with Solution

Given:

• $\overrightarrow{a} = i + 2j + k$
• $\overrightarrow{b} = 2i + 7j + 3k$
• $L_1 : \overrightarrow{r} = (-i + 2j + k) + \lambda \overrightarrow{a}$
• $L_2 : \overrightarrow{r} = (j + k) + \mu \overrightarrow{b}$

Find: The point through which $L_3$ passes, where $L_3$ goes through the intersection of $L_1$ and $L_2$ and is parallel to $\overrightarrow{a} + \overrightarrow{b}$.

To find the intersection of the two lines, equate their vector forms:

$$(-\hat{i} + 2\hat{j} + \hat{k}) + \lambda (\hat{i} + 2\hat{j} + \hat{k}) = (\hat{j} + \hat{k}) + \mu (2\hat{i} + 7\hat{j} + 3\hat{k})$$

Comparing coefficients of $\hat{i}$, $\hat{j}$ and $\hat{k}$:

$$\begin{aligned} -1 + \lambda &= 2\mu \\ 2 + 2\lambda &= 1 + 7\mu \\ 1 + \lambda &= 1 + 3\mu \end{aligned}$$

From the third equation,

$$\lambda = 3\mu$$

Substitute in the first equation:

$$-1 + 3\mu = 2\mu$$

So,

$$\mu = 1$$

Hence,

$$\lambda = 3$$

Check in the second equation:

$$2 + 2(3) = 1 + 7(1)$$ $$8 = 8$$

Therefore, the lines intersect at the point obtained from $L_1$ with $\lambda = 3$:

$$\overrightarrow{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + 3(\hat{i} + 2\hat{j} + \hat{k})$$ $$\overrightarrow{r} = 2\hat{i} + 8\hat{j} + 4\hat{k}$$

So the intersection point is $(2, 8, 4)$.

Now find the direction vector of $L_3$:

$$\overrightarrow{a} + \overrightarrow{b} = (\hat{i} + 2\hat{j} + \hat{k}) + (2\hat{i} + 7\hat{j} + 3\hat{k}) = 3\hat{i} + 9\hat{j} + 4\hat{k}$$

Hence,

$$L_3 : \overrightarrow{r} = (2\hat{i} + 8\hat{j} + 4\hat{k}) + t(3\hat{i} + 9\hat{j} + 4\hat{k}), \quad t \in \mathbb{R}$$

This gives the parametric coordinates:

$$\begin{aligned} x &= 2 + 3t \\ y &= 8 + 9t \\ z &= 4 + 4t \end{aligned}$$

Now test the options. For option $A$, the point is $(-1,-1,1)$. Using the $x$-coordinate:

$$-1 = 2 + 3t \Rightarrow t = -1$$

Check the other coordinates at $t = -1$:

$$y = 8 + 9(-1) = -1, \qquad z = 4 + 4(-1) = 0$$

So option $A$ does not satisfy the third coordinate.

There is a discrepancy in the solution because it concludes with the point $(5,17,8)$, which is not present in the options, while the page also marks option $A$ as correct. Among the given options, the intended answer from the provided answer key is A.

Therefore, the correct option is A according to the provided source.