Chapter 10.4, Problem 73AYU

In Problems 67-74, analyze each equation.

$x^2-6x-8y-31=0$

To determine

To find: Analyze the equation.

Answer to Problem 73AYU

The equation represents parabola.

Explanation of Solution

Given:

$x^2-6x-8y-31=0$

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(h,k\right)$	$\left(0,k+a\right)$	$y=k-a$	${\left(x-h\right)}^2=4a\left(y-k\right)$	Parabola,axis of symmetry is parallel to $y\text{-axis}$, opens up

Calculation:

Given that $x^2-6x-8y-31=0$.

${\left(x-3\right)}^2-8\left(y+4\right)=9+31-32$

${\left(x-3\right)}^2=8\left(y+5\right)$

The equation represents parabola.

The axis of symmetry is parallel to $y\text{-axis}$.

Vertex: $V\left(h,k\right)=\left(3,-\text{5}\right)$.

$4a=8$, $a=2$

Focus: $F\left(h,k+a\right)=F\left(3,-3\right)=F\left(3,-3\right)$.

The coordinates of the focus is $F\left(3,-3\right)$.

Directrix: $y=-5-2=-7$.

The graph of the equation of $x^2-6x-8y-31=0$ is plotted.

Using the given equation $x^2-6x-8y-31=0$, we plot the graph using graphing tool.

The graph represents a parabola, with vertex $\left(3,-5\right)$, directrix $y=-7$ and the axis of symmetry is parallel to $y\text{-axis}$, opens up.