Chapter 10, Problem 6RE

In Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes.

$x^2-4x\text{ = }2y$

To determine

Each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its center, vertices, foci, and asymptotes.

Answer to Problem 6RE

Parabola

Vertex: $\left(2,-2\right)$

Focus: $\left(2,-\frac{3}{2}\right)$

Directrix: $y=-\frac{5}{2}$.

Explanation of Solution

Given:

$x²-4x=2y$

Formula used:

Vertex	Focus	Directrix	Equation
$\left(h,k\right)$	$\left(h,k+a\right)$	$y=k-a$	$\left(x-h\right)²=4a\left(y-k\right)$

Calculation:

The equation $x²-4x=2y$ can be written as ${\left(x-2\right)}^2=2\left(y+2\right)$.

The above equation is of the form $\left(x-h\right)²=4a\left(y-k\right)$ and it represents a parabola.

$a=1/2,h=2,k=-2$

Vertex: $\left(h,k\right)=\left(2,-2\right)$.

Focus: $\left(h,k+a\right)=(2,-2+1/2)=\left(2,-\frac{3}{2}\right)$.

Directrix: $y=k+a=-5/2$.