Question

The focal distance of a point on the parabola is $y^2=8x\text{ is 4}$; Find the coordinates of the point?

Answer 1

Hint – First of all, read the question carefully and write the things given in the question i.e. focal distance of the parabola is $4$ i.e. $\text{D = 4}$ which is the distance between the focus and that particular point on the parabola and let the given point and coordinates be $\text{P}\left(x,y\right)$. The equation of parabola i.e. $y^2=8x$. Now, this will give us a clear picture to understand the question. Thus we will get our desired answer.

“Complete step-by-step answer:”
 Now, we will find the coordinates of that particular point. We will use the standard parabola equation i.e. $y^2=4ax$ to solve this given problem.
So, compare the given equation $y^2=8x$ with the standard equation of parabola i.e. $y^2=4ax$, then we will find that $\text{a = 2}$ by comparing $8x\text{ and }4ax$.
As we know that the standard focus of the parabola is $\left(a,0\right)$. Hence, the focus $\text{F}$ of the given parabola is $\left(2,0\right)$.
According to the question $\text{D = 4}$ and we assumed the point on the locus as $\text{P}\left(x,y\right)$.
By using the distance the formula on $\text{F}\left(2,0\right)\text{ and P}\left(x,y\right)$ we will get ,
$\text{4 = }\sqrt{{\left(x-2\right)}^2+{\left(y-0\right)}^2}$
By squaring on both sides,
$16={\left(x-2\right)}^2+{\left(y\right)}^2$
Now, put the value of $y^2$ as $8x$ which is given in question and expand the equation ${\left(x-2\right)}^2$
$16=x^2+4-4x+8x$
$x^2+4x-12=0$
By using factorisation method,
$x^2+6x-2x-12=0$
$x\left(x+6\right)-2\left(x+6\right)=0$
$\left(x-2\right)\left(x+6\right)=0$
This implies that $x$ can be $2,-6$ but according to the equation $y^2=4ax$, $x$ cannot be negative.
So, we left with only $x=2$
Now, by putting the value of $x$ in $y^2=8x$
We will get,
$y^2=8\times 2$
$y^2=16$
Apply square root both sides, we will get
$y=4,-4$
So, the coordinates are $\left(2,4\right)\text{ and }\left(2,-4\right)$

Note – In this type of questions, firstly we should compare the given equation with the standard parabolic equations which are:
$$\begin{gathered} \left(1\right)y^2\text{ = 4}ax \\ \left(2\right){\text{y}}^2=-4ax \\ \left(3\right)x^2=4ay \\ \left(4\right)x^2=-4ay \end{gathered}$$
Then simply putting those values in the equation we get our required answer.
Do note that the distance formula between the two points i.e. $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ is:
$\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}=\text{ Distance between them }$ .