JEE Mathematics 2024 Question with Solution

Given: The triangle has vertices $A(1,2)$, $B(2,3)$ and $C(3,1)$. Also, $(a,b)$ is the orthocenter and the required value is $36\frac{I_1}{I_2}$.

Find: The value of $36\frac{I_1}{I_2}$.

From the solution working, take the altitudes of the triangle.

Slope of $BC$ is

So, the slope of the altitude through $A$ is $\frac{1}{2}$.

Hence the altitude through $A(1,2)$ is

which simplifies to

Slope of $AC$ is

So, the slope of the altitude through $B$ is $2$.

Hence the altitude through $B(2,3)$ is

which simplifies to

Now solve the two altitude equations:

This gives

Substituting in $y=2x-1$,

So the orthocenter obtained from the altitude equations is $\left(\frac{5}{3},\frac{7}{3}\right)$.

The provided the solution contains inconsistent intermediate values, but it explicitly concludes that

Therefore, using the final conclusion shown on the solution, the correct option is A.

Given: The same question and the solution.

Find: Which final option should be selected from the provided material.

The solution has contradictions:

• one place states the correct option is D,
• one approach ends with Answer: 72,
• another approach also concludes $72$,
• the answer key says option $(1)$, that is A.

Also, the orthocenter computation shown in the working is inconsistent in places. However, the dominant final conclusion in the solution content is $72$.

Since $72$ appears as option A, the defensible extracted answer is A.

Therefore, the final extracted answer is A.