JEE Mathematics 2023 Question with Solution

Given: Two sides of the triangle are $x = 0$ and $y = 3$. The third side is a tangent to the parabola $y^2 = 6x$.

Find: The locus of the circumcentre of the triangle.

For the parabola $y^2 = 4ax$, we have

so

A tangent to $y^2 = 4ax$ in slope form is

Hence here the tangent is

with $m \ne 0$.

This tangent meets $x = 0$ at

and meets $y = 3$ at

Since the two given sides are perpendicular, the triangle is right-angled at $\left(0,3\right)$. Therefore, its circumcentre is the midpoint of the hypotenuse.

So if the circumcentre is $\left(h,k\right)$, then

Now eliminate $m$ using the given relation from the solution:

which gives

Therefore,

Replacing $\left(h,k\right)$ by $\left(x,y\right)$, the locus is

Therefore, the correct option is C.

Given: The triangle is formed by the lines $x = 0$, $y = 3$, and a tangent to $y^2 = 6x$.

Find: The locus of the circumcentre.

The intersection of $x = 0$ and $y = 3$ is $\left(0,3\right)$, so the angle between these two sides is a right angle.

Let the tangent be

Then its intercept on $x = 0$ is

and its intercept on $y = 3$ is found from

so

and hence

Thus

The hypotenuse is $AB$, so the circumcentre is its midpoint:

Eliminating $m$ gives

So the locus of the circumcentre is the parabola represented by this equation.

Therefore, the correct option is C.