Question

Find the equation of parabola with vertex (0, 0) and focus at (0, 2).

Answer 1

Hint: First find the type of parabola possible with these two points. Now apply the formula to get the required equation, by these formulas you can get a variable which is the only variable needed to define a parabola at origin. This equation of parabola is the required result.

Complete step-by-step answer:
Parabola: In mathematics, parabola is a plane curve which is mirror symmetrical, approximately U-shape. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves. It is defined as the locus of the point which is equidistant from a fixed point focus, fixed line directrix.
Given points of parabola can be written as in the form of vertex of parabola origin (0, 0). Focus of parabola is (0, 2). Given focus lies on the y-axis, the parabola also faces towards the y-axis. A parabola facing y-axis have 2 possibilities:
$\begin{align}
  & {{\left( x-h \right)}^{2}}=4a\left( y-k \right) \\
 & {{\left( x-h \right)}^{2}}=-4a\left( y-k \right) \\
\end{align}$
Here vertex is origin, so we get values of h, k as 0, 0 respectively. By substituting these values into equations possible we get
$\begin{align}
  & {{\left( x-0 \right)}^{2}}=4a\left( y-0 \right) \\
 & {{\left( x-0 \right)}^{2}}=4a\left( y-0 \right) \\
\end{align}$
By simplifying the above equation, we can write them as
$\begin{align}
  & {{x}^{2}}=4ay \\
 & {{x}^{2}}=-4ay \\
\end{align}$
Given focus is (0, 2) it lies on the positive y-axis. So, the parabola must be facing towards positive y-axis, by the above statement we get the required equation as,
${{x}^{2}}=4ay$
We need the value of ‘a’ to get the equation of parabola, we know the focus of parabola is at a distance from the vertex. So, here vertex is (0, 0) focus is (0, 2) so distance between them is. Distance between (a, b);(c, d) is
$d=\sqrt{{{\left( a-c \right)}^{2}}+{{\left( b-d \right)}^{2}}}$
Here we gave a = b = c = 0, d = 2. So distance is $d=\sqrt{{{2}^{2}}}=2$

 By above statement, we can write equation of ‘a’ as:
a = distance between (0, 0), (0, 2)
By substituting their values, we get the value of ‘a’ as a = 2. By substituting in equation of parabola we get it as
${{x}^{2}}=4\left( 2y \right)=8y$
So, equation is
${{x}^{2}}=8y$
Therefore this is required parabola.

Note: Always follow this step by step process of eliminating possibilities to get the correct result, negative root is not taken in distance because distance is always positive. Some have misconceptions like distance between focus and vertex is 4a, but it is wrong the distance is always “a”.