Question

find the vertex and focus of the parabola: x² + 14x - 8y + 65 = 0 vertex = ( ?, ) focus = (, )

Explanation:

Step1: Rewrite the given equation in standard form

First, complete the square for the $x$ - terms.

$$
LATEXBLOCK0
$$

Step2: Identify the vertex

The standard form of a parabola opening upwards is $(x - h)^{2}=4p(y - k)$, where $(h,k)$ is the vertex.
Comparing $(x + 7)^{2}=8(y - 2)$ with $(x - h)^{2}=4p(y - k)$, we have $h=-7$, $k = 2$. So the vertex is $(-7,2)$.

Step3: Find the value of $p$

Since $(x + 7)^{2}=8(y - 2)$ and $(x - h)^{2}=4p(y - k)$, then $4p = 8$, so $p = 2$.

Step4: Identify the focus

The focus of a parabola of the form $(x - h)^{2}=4p(y - k)$ is given by the point $(h,k + p)$.
Substituting $h=-7$, $k = 2$, and $p = 2$ into the formula for the focus, we get the focus as $(-7,2 + 2)=(-7,4)$.

Answer:

Vertex = $(-7,2)$
Focus = $(-7,4)$