JEE Mathematics 2024 Question with Solution

Given:

• $A = (2,2,1)$
• $B = (1,2,2)$
• $C = (2,1,2)$

Find: $l_1^2 + l_2^2 + l_3^2$, where $l_1, l_2, l_3$ are the perpendicular distances from the orthocenter to sides AB, BC, and CA.

First compute the side vectors:

Their lengths are equal:

Hence, $\triangle ABC$ is equilateral.

For an equilateral triangle, the orthocenter coincides with the centroid. Therefore the orthocenter is

Now find the perpendicular distance from $G$ to side AB. Let $D$ be the midpoint of AB:

In an equilateral triangle, the median is also the perpendicular from the center to the side, so

Compute $GD$:

Since the triangle is equilateral,

Therefore,

Therefore, the correct option is B, and the value is $\frac{1}{2}$.

Given: the vertices are $A(2,2,1)$, $B(1,2,2)$, and $C(2,1,2)$.

Find: $l_1^2 + l_2^2 + l_3^2$.

Check the three side lengths directly:

So the triangle is equilateral. Hence its orthocenter, centroid, and incenter are the same point.

In an equilateral triangle of side $a$, the distance from the center to any side is the inradius:

Here $a = \sqrt{2}$, so

Thus,

Now square and add:

Therefore, the correct option is B.