Question

The focal distance of a point on the parabola ${{y}^{2}}=12x$ is 4. Find the abscissa of this point.

Answer 1

Hint: A parabola is given ${{y}^{2}}=12x$, compare it with the general equation of parabola, ${{y}^{2}}=4ax$. You will get the value of a. Now focal distance is equal to $x+a$. So, you will get an abscissa.

Complete step by step answer:
A parabola is a curve that looks like the one shown above. Its open end can point up, down, left or right. A curve of this shape is called 'parabolic', meaning 'like a parabola'.
A U-shaped curve with certain specific properties, formally it can be defined as for the given point, called as focus and given line not through the focus, called the directrix, a parabola is locus of points such that distance to the focus equals the distance to the directrix.
The vertex and the focus determine a line, perpendicular to the directrix that is the axis of the parabola. The line through the focus parallel to the directrix is the latus rectum (straight side). The parabola is symmetric about its axis, moving farther from the axis as the curve recedes in the direction away from its vertex.
The focal distance is the distance from the origin to the focus and from the origin to the directrix. We take absolute value because distance is positive.
Now, we want to find abscissa.
Let $x$ be the abscissa of the point.
So, we have been given the focal distance as 4.
We know the general equation of parabola, i.e.,
${{y}^{2}}=4ax$
Comparing the general equation of parabola, ${{y}^{2}}=4ax$, with given equation, ${{y}^{2}}=12x$, we get the value of $a=3$.
We know that the focal distance is, $=x+a$
So, the focal distance is 4 and $a=3$.
So, we get the entire equation as,
 $x+a=4$
Simplifying the above equation by substituting the value of ‘a’, we get,
$x+3=4$
Subtracting 3 from both sides, we get the answer as,
$x+3-3=4-3$
Simplifying further, we get,
$x=1$
So, we get the value of abscissa as 1.

Note: Read the question carefully. You should be knowing focal distance and concepts of parabola i.e. focus directrix etc. You should know that the focal distance of ${{y}^{2}}=4ax$ which is equal to $x+a$. The focal distance is the distance from the origin to the focus and from the origin to the directrix. We take absolute value because distance always has to be positive.