Question

find the vertex and focus of the parabola: x² - 10x - 12y - 23 = 0 vertex = (?, ) focus = (, )

Explanation:

Step1: Rewrite the 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 or downwards is $(x - h)^{2}=4p(y - k)$, where $(h,k)$ is the vertex.
Comparing $(x - 5)^{2}=12(y + 4)$ with $(x - h)^{2}=4p(y - k)$, we have $h = 5$ and $k=-4$. So the vertex is $(5,-4)$.

Step3: Identify the value of $p$

Since $(x - 5)^{2}=12(y + 4)$ and $(x - h)^{2}=4p(y - k)$, then $4p = 12$, so $p = 3$.

Step4: Find the focus

For a parabola of the form $(x - h)^{2}=4p(y - k)$ opening upwards, the focus is given by the point $(h,k + p)$.
Here, $h = 5$, $k=-4$, and $p = 3$. So the focus is $(5,-4 + 3)=(5,-1)$.

Answer:

Vertex = $(5,-4)$
Focus = $(5,-1)$