Chapter 10.2, Problem 77AYU

Show that an equation of the form

$C{y}^2\text{ + }Dx\text{ = 0}$     $C\ne \text{ 0}$, $D\ne \text{ 0}$

is the equation of a parabola with vertex at $\left(0,\text{ 0}\right)$ and axis of symmetry the $x\text{-axis}$. Find its focus and directrix.

To determine

To find:

a.An equation of the form $Cy²+Dx=0$; $C\ne 0,D\ne 0$ is the equation of a parabola with vertex at $\left(0,0\right)$ and axis of symmetry the $x\text{-axis}$.

Answer to Problem 77AYU

$y^2=-\frac{D}{C}x$

Explanation of Solution

Given:

$Cy²+Dx=0$; $C\ne 0,D\ne 0$

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(0,0\right)$	$\left(a,0\right)$	$x=-a$	$y²=4ax$	Parabola, axis of symmetry is $x\text{-axis}$, opens right
$\left(0,0\right)$	$\left(-a,0\right)$	$x=a$	$y²=-4ax$	Parabola, axis of symmetry is $x\text{-axis}$, opens left

Calculation:

$C{y}^2+Dx=0$

$Cy²=-Dx$

$y^2=-\frac{D}{C}x$

$C{y}^2+Dx=0$; $C\ne 0,D\ne 0$ is the equation of a parabola with vertex $\left(0,0\right)$ and axis of symmetry is $x\text{-axis}$.