MAH-CET 2026 — Mathematics PYQ

Step 1: Identify properties of the standard parabola and focal chord

The given parabola is:

$$y^2=4ax$$

For this standard parabola:

• The coordinates of its focus $S$ are $(a,0)$.

• A focal chord is any chord that passes through the focus $S(a,0)$.

Step 2: Use the equation of a chord with a given midpoint

Let the midpoint of the focal chord be $P(h,k)$. We need to find the locus of this point $P$.

The equation of a chord of a parabola $y^2=4ax$ whose midpoint is $(h,k)$ is given by the standard formula:

$$T=S_1$$

Where:

• $T=yk-2a(x+h)$

• $S_1=k^2-4ah$

Equating $T=S_1$:

$$yk-2a(x+h)=k^2-4ah$$

$$yk-2ax-2ah=k^2-4ah$$

$$yk-2ax=k^2-2ah$$

Step 3: Apply the condition that the chord passes through the focus

Since this chord is a focal chord, it must pass through the focus $S(a,0)$.

Substitute $x=a$ and $y=0$ into the chord equation:

$$(0)k-2a(a)=k^2-2ah$$

$$-2a^2=k^2-2ah$$

Rearranging the equation to solve cleanly:

$$k^2=2ah-2a^2$$

$$k^2=2a(h-a)$$

Step 4: Find the locus

To find the final locus equation, replace the arbitrary midpoint coordinates $h$ with $x$ and $k$ with $y$:

$$y^2=2a(x-a)$$

Correct Answer

(a) $y^2=2a(x-a)$