JEE Mathematics 2024 Question with Solution

Given: $R(-1,4,2)$ and $M(2,1,2)$ is the midpoint of $PQ$.

Find: The distance between the centroid of $\triangle PQR$ and the point of intersection of the two given lines.

Use the fact that the centroid of a triangle lies on the median joining a vertex to the midpoint of the opposite side and divides that median in the ratio $2:1$ from the vertex. Thus, if $M$ is the midpoint of $PQ$, then the centroid $G$ lies on $RM$ and divides it internally in the ratio $2:1$.

So,

Now find the intersection point of the lines.

For the first line,

For the second line,

At the point of intersection, coordinates are equal. From $x$,

Then from the second line,

Hence the intersection point is

Now use the distance formula between $G(1,2,2)$ and $A(2,-6,0)$:

Therefore, the distance is $\sqrt{69}$ and the correct option is C.

Given: $M$ is the midpoint of $PQ$ and $R(-1,4,2)$ is the third vertex.

Find: The required distance.

Instead of introducing coordinates of $P$ and $Q$, directly use the centroid property on median $RM$. The centroid is the section point that divides $RM$ internally in the ratio $2:1$ from $R$ toward $M$.

Thus,

The line intersection is obtained quickly by noticing that the first line has $x=2$ always. On the second line, set $1+\mu=2$, giving $\mu=1$. Hence the intersection point is

Now compute

Therefore,

Hence the correct option is C.