JEE Mathematics 2023 Question with Solution

Given: The rectangle is shown with intercepts leading to points $A(2h,0)$ and $B(0,2k)$, and the line segment $AB$ divides the rectangle in the ratio $4:1$.

Find: The locus of the midpoint of $AB$.

From the solution,

Also,

So,

If the midpoint is $M(h,k)$, then its coordinates satisfy

Hence, the midpoint lies on a hyperbola.

Therefore, the correct curve is Hyperbola. The solution states the correct option is B, although the listed option text shows Hyperbola as option D. the answer is recorded as B.

Given: $A(\alpha,0)$ and $B(0,\beta)$, and the midpoint of $AB$ is

Find: The nature of the locus of $M$.

The extracted solution states that the midpoint has coordinates

and concludes that this locus is a hyperbola.

Using the area relation written in the extracted solution, the product of the midpoint coordinates becomes constant, which is of the form

for some constant $c$. Such an equation represents a hyperbola.

Therefore, the midpoint of $AB$ lies on a hyperbola.