JEE Mathematics 2024 Question with Solution

Given: The points $M$ and $N$ lie on the lines

and

with $MN$ as the shortest distance between the lines.

Find: $OM \cdot ON$

Parametrize the lines:

Then

The direction vectors are

Their cross product is

Since $MN$ is the shortest distance between the skew lines, $\overrightarrow{MN}$ is parallel to $\overrightarrow{b_1} \times \overrightarrow{b_2}$. Hence,

So we get

and

Solving these equations,

Therefore,

So,

Now compute the dot product:

Therefore, the value of $OM \cdot ON$ is $9$.

Given: Points $M$ and $N$ lie on the two given lines, and $MN$ is the shortest distance between them.

Find: $OM \cdot ON$

For the first line, let the common ratio be $\lambda$. Then

which gives

Hence,

For the second line, let the common ratio be $\mu$. Then

which gives

Hence,

Now,

Using the coordinates listed in the extracted solution, this is written as

The line directions are

Then

For shortest distance between two skew lines, the joining vector is parallel to this cross product. Therefore the corresponding component relations give

and

Subtracting the second equation from the first,

so

Substitute into

to get

Now substitute back:

Therefore,

So the required numerical value is $9$.