Question

How do you find the standard equation given focus (8,10), and vertex (8,6)?

Answer 1

Use the vertex form of the equation and use the focus to compute the value of ‘a’ and then substitute these values into the vertex form equation. The first thing we must do here is to try to find the focal distance, which is the distance from the vertex to the focus and then we compute the value of ‘a’ by using the formula $a = 1/4f$. After doing all this we will then substitute the values of ‘a’ and vertex points into the vertex form of the equation which is given by and expand it which will finally yield our answer.

Complete step by step solution:
We first find the focal distance $f$, which is the distance from the vertex to the focus point.
$\Rightarrow f = 10 - 6$
$\Rightarrow f = 4$
We will now compute the value of ‘a’
The value of ‘a’ is given by $a = \dfrac{1}{4}f$.
Substituting the value of $f$ in the above equation to find ‘a’ we get,
$\Rightarrow a = \dfrac{1}{{16}}$
Now the vertex point shows that $h = 8$ and $k = 6$
Now substituting these and values into the vertex form of the equation:
$\Rightarrow y = a{(x - h)^2} + k$
$\Rightarrow y = \dfrac{1}{{16}}{(x - 8)^2} + 6$
Now expanding the square:
$\Rightarrow y = \dfrac{1}{{16}}({x^2} - 16x + 64) + 6$
$\Rightarrow y = \dfrac{{{x^2}}}{{16}} - x + 4 + 6$
Therefore the standard equation is
$\Rightarrow y = \dfrac{{{x^2}}}{{16}} - x + 10$

Note: Here we must understand that the focus is above the vertex and therefore the above general vertex form of the equation is used. And while substituting we must ensure that we must substitute the vertex points and not the focus point. Also after substituting it is important to expand the equation to the standard form.