Answer
The prove is as below.
Work Step by Step
Here, the point $P$ is on the parabola $y=x^2$, so, the coordinate will be $(x, x^2)$.
Now, the distance between point $\left(0, \dfrac{1}{4}\right)$ and $P(x, x^2)$ can be calculated using distance formula as follows:
Distance, $d_{1}=\sqrt{(x-0)^2+\left(x^2-\dfrac{1}{4}\right)^2}$
$\Rightarrow d_{1}=\sqrt{x^2+(x^2)^2-2(x^2)\left(\dfrac{1}{4}\right)+\left(\dfrac{1}{4}\right)^2}$
$\Rightarrow d_{1}=\sqrt{x^2+(x^2)^2-\dfrac{x^2}{2}+\left(\dfrac{1}{4}\right)^2}$
$\Rightarrow d_{1}=\sqrt{(x^2)^2+\dfrac{x^2}{2}+\left(\dfrac{1}{4}\right)^2}$
$\Rightarrow d_{1}=\sqrt{(x^2)^2+2(x^2)\left(\dfrac{1}{4}\right)+\left(\dfrac{1}{4}\right)^2}$
$\Rightarrow d_{1}=\sqrt{\left(x^2+\dfrac{1}{4}\right)^2}$
$\Rightarrow d_{1}=x^2+\dfrac{1}{4}$ units.
Further, point $P(x, x^2)$ is $x^2$ units above x-axis and the line $y=-\dfrac{1}{4}$ is $\dfrac{1}{4}$ units below x-axis, so, the distance between point $P$ and line $y=-\dfrac{1}{4}$ is $d_{2}=x^2+\dfrac{1}{4}$ units.
Therefore, we have $d_{1}=d_{2}=x^2+\dfrac{1}{4}$. Hence, proved.
