NIMCET 2023 — Mathematics PYQ

1. Identify Given Properties

• Given Parabola: $y^2=4x$

• The vertex of this standard parabola is at the origin: $O(0,0)$.

• Let $P$ be any moving point on the parabola. The parametric coordinates of any point on $y^2=4ax$ are $(a{t}^2,2at)$.

• Comparing $y^2=4x$ with $y^2=4ax$, we get $a=1$. Thus, the coordinates of point $P$ are:

  $$P=(t^2,2t)$$

2. Define the Mid-point Variable

Let $OP$ be the chord passing through the vertex $O(0,0)$ and point $P(t^2,2t)$. Let $M(h,k)$ be the mid-point of this chord $OP$.

Using the midpoint section formula:

$$h=\frac{0+t^2}{2}⟹h=\frac{t^2}{2}$$

$$k=\frac{0+2t}{2}⟹k=t$$

3. Eliminate Parameter $t$

Substitute the value of $t=k$ into the expression for $h$:

$$h=\frac{(k{)}^2}{2}$$

$$2h=k^2⟹k^2=2h$$

4. Form the Final Locus Equation

To write down the final equation for the locus, replace $h$ with $x$ and $k$ with $y$:

$$y^2=2x$$