Question

The focal distance of a point on the parabola \[{{y}^{2}}=12x\] is 4. Find the abscissa of this point.
A. 1
B. -1
C. 3
D. None of these

Answer 1

Hint: For a general parabola which is denoted as \[{{y}^{2}}=4ax\], the focus is located at (a, 0).
For calculating the focal distance, the formula for a point \[({{x}_{1}},{{y}_{1}})\]
\[d=\sqrt{{{({{x}_{1}}-a)}^{2}}+{{({{y}_{1}})}^{2}}}\]
For a parabola, the distance between a point and the focus is equal to the distance between the point and the directrix of the parabola.
Distance between a point and a line is
\[d=\dfrac{a{{x}_{1}}+b{{y}_{1}}+c}{\sqrt{{{a}^{2}}+{{b}^{2}}}}\]

Complete step-by-step answer:

As mentioned in the question and also by using the formula of a general parabola, we get
 a = 3 (As when we make the given equation of the parabola and compare it with the general formula, we get the value of a as 3)
Now, we know the formula for calculating the focal distance as given the hint as well as
\[d=\sqrt{{{({{x}_{1}}-3)}^{2}}+{{({{y}_{1}})}^{2}}}\]
Now, we know that the value of d is 4 as given in the question itself, so further we can write the above equation as
\[4=\sqrt{{{({{x}_{1}}-a)}^{2}}+{{({{y}_{1}})}^{2}}}\]
Now, on squaring both the side we get
\[\begin{align}
  & {{4}^{2}}={{({{x}_{1}}-3)}^{2}}+{{({{y}_{1}})}^{2}} \\
 & 16={{({{x}_{1}}-3)}^{2}}+{{({{y}_{1}})}^{2}} \\
\end{align}\]
Now, we know that for a parabola, the distance between a point and the focus is equal to the distance between the point and the directrix of the parabola.
So, the equation of the directrix of a general parabola can be written as
\[\begin{align}
  & x=-a \\
 & x=-3 \\
\end{align}\]
Now, the distance between the point and the directrix is
\[\begin{align}
  & d=\dfrac{1{{x}_{1}}+0{{y}_{1}}+3}{\sqrt{{{1}^{2}}+{{0}^{2}}}} \\
 & 4={{x}_{1}}+3 \\
 & {{x}_{1}}=1 \\
\end{align}\]
Hence, the abscissa of that point is +1.

NOTE: The students can get the question wrong if they do not know the trick that for a parabola, the distance between a point and the focus is equal to the distance between the point and the directrix of the parabola. This information is very important as this question is tailor made for this fact only.