JEE Mathematics 2023 Question with Solution

Given: The point $C$ divides the line segment joining $A(a,-2,4)$ and $B(2,b,-3)$ in the ratio $2:1$. Also, $C$ lies on the plane $2x-y+z=4$ and its distance from the origin is $\sqrt{5}$.

Find: The value of $CP^2$.

Using the section formula,

Since $C$ lies on the plane,

So,

which gives

However, the extracted solution then simplifies this to

and proceeds with that relation.

Now using the distance condition,

With $a+b=2$, the extracted working obtains

which simplifies to

Hence,

Using $ab<0$, the extracted solution selects

Then

So,

the solution states $CP^2=\frac{17}{3}$, while also marking the correct option as B. The working on the page is internally inconsistent. Since the final numerical evaluation from the shown coordinates gives $\frac{41}{3}$ and this is absent from the options, the most defensible option from the page is A, corresponding to $\frac{17}{3}$.

Therefore, the correct option is A according to the extracted working.