Question

How do you write an equation of a parabola with its vertex at the origin and focus at \[\left( 0,-5 \right)\].

Answer 1

Hint: In this problem, we have to find the equation of the parabola whose vertex is at the origin and focus is \[\left( 0,-5 \right)\]. we know that the parabola is open downwards and the focus is below the vertex point. So, we have to write the general equation of the open downwards parabola. We also know that the distance between focus point and the vertex point is a, finding and substituting in the equation, we can get the equation of parabola.

Complete step by step answer:
We know that the general equation of the open downward parabola is,
\[{{\left( x-h \right)}^{2}}=-4a\left( y-k \right)\] ……. (1)
We also know that the given vertex is the origin \[\left( 0,0 \right)\] which is \[\left( h,k \right)\].
Now we can substitute the origin point in the equation (1).
\[{{x}^{2}}=-4ay\]…….. (2)
We also know that the distance between focus and vertex is a.
The distance between \[\left( 0,0 \right)\] and \[\left( 0,-5 \right)\] is 5
Therefore, a is 5.
Now we can substitute the value of a in equation (2)
\[\begin{align}
  & \Rightarrow {{x}^{2}}=-4\left( 5 \right)y \\
 & \Rightarrow {{x}^{2}}=-20y \\
\end{align}\]

Therefore, the equation of the parabola \[{{x}^{2}}=-20y\].

Note: Students make mistakes while finding the distance between the focus point and the vertex point, which must be concentrated. We can use the distance formula to find the distance between the focus and the vertex point. We should also substitute the vertex point in the general equation.