Question

Find the equation of parabola with focus (5,0) and vertex (5,3).

Answer 1

Hint: The given question is of parabola, in which we need to obtain the equation of parabola, here first we need to decide the type of parabola and then get the equation of parabola and thus solve it further.

Formula used:
The equation of parabola, when the vertex is below the focus:
\[ \Rightarrow {(x - h)^2} = 4p(y - k)\]

Complete answer:
Here first we see that the vertex is below the focus, hence it is the right side up parabola and “p” is positive.

Now equation of parabola:
\[ \Rightarrow {(x - h)^2} = 4p(y - k)\]
Focus : (5,0)
Vertex : (5,3)
\[ \Rightarrow p = 0 - 3 = - 3\]
Therefore, putting value in the equation we get:
\[ \Rightarrow {(x - 5)^2} = 4( - 3)(y - 3)\]
Now on solving the equation of parabola is:
\[
   \Rightarrow \dfrac{{ - {{(x - 5)}^2}}}{{12}} = y - 3 \\
   \Rightarrow \left( {\dfrac{{ - {{(x - 5)}^2}}}{{12}}} \right) + 3 = y \\
 \]
Here we obtained the final equation of parabola.

Additional Information: Here the equation of parabola for the right side up parabola is chosen according to the coordinates and vertex given. Hence further we put the value of “p” after solving and then obtain the equation of parabola.

Note:
To solve the equation of parabola, one needs to solve the equation of parabola and then put the values obtained from the vertex and focus coordinates given. Then further the solution needs to be solved accordingly.