JEE Mathematics 2025 Question with Solution

Given: The line $L$ passes through $C(1,1,1)$ and intersects the two given lines.

Find: Which given point lies on the line $L$.

Take a point on the second line as $A(\mu+3,\, 2\mu+4,\, \mu)$ and a point on the first line as $B(2\lambda+1,\, 3\lambda-1,\, 4\lambda+1)$.

Since $A, B, C$ lie on the required line $L$, the direction ratios of $AC$ and $BC$ must be proportional.

For collinearity,

Using the first and third ratios,

Substituting $\mu=-5$ in the direction ratios of $AC$,

So a direction ratio of line $L$ is $(3,7,6)$.

Hence the equation of the required line is

Now check option $D$, $(7,15,13)$:

All three ratios are equal, so the point lies on $L$.

Therefore, the point lying on the line $L$ is $(7,15,13)$. The correct option is D.

Given: $L$ passes through $(1,1,1)$ and intersects both given lines.

Find: The point among the options that lies on $L$.

Write the given lines in parametric form:

For $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} = \lambda$,

For $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1} = \mu$,

So two general points on the lines are

and

Since $A, B, C(1,1,1)$ are collinear, their connecting direction ratios must be proportional:

Thus,

From

we get

Therefore,

So the required line has direction ratios proportional to $(3,7,6)$ and passes through $(1,1,1)$.

Hence,

Testing the options, only $(7,15,13)$ satisfies

Therefore, the correct option is D.