Chapter 1.2, Problem 86AYU

To determine

To find: The intercepts of the graph of the equation $16y^2\text{ = 120}x\text{ − 225}$ and symmetry with respect to $x\text{-axis}$, $y\text{-axis}$ and the origin.

Answer to Problem 86AYU

The $x\text{-intercept}$ is $\text{1}\text{.875}$

The graph of the equation is symmetric with respect to $\mathrm{x-axis}$

Explanation of Solution

Given:

The equation $16y^2\text{ = 120x − 225}$

Calculation:

To find the $x\text{-intercept}$ let $y\text{ = 0}$ in the equation

$16y^2\text{ = 120}x\text{ − 225}$

$16{(0)}^2\text{ = 120}x\text{ − 225}$

$\text{0 = }\text{120}x-225$

$0+225\text{ = 120}x\text{ <mo>−</mo> 225 + 225}$

$225\text{ = 120}x$

$\text{120}x\text{ = 225}$

$x\text{ = 1}\text{.875}$

The $x\text{-intercept}$ is $\text{1}\text{.875}$

To find the $y\text{-intercept}$ let $x\text{ = 0}$ in the equation

$16y^2\text{ = 120}x - \text{225}$

$16y^2\text{ = 120(0) − 225}$

$16y^2\text{ = }\mathrm{-225}$

$y^2\text{ = }\frac{\mathrm{-225}}{120}$   Divide the equation by 120

$y^2\text{ = }-1.875$

$y\text{ = }\sqrt{-1.875}$   not real

There is no $y\text{-intercept}$

To test the symmetry of the graph with respect to $x\text{-axis}$, replace $y$ by $\text{−}y$ in the equation

$16{(-y)}^2=\text{ 120 }x-\text{ 225}$

$16y^2=\text{ 120}x\text{ - 225}$   which is equivalent to the original equation.

The graph of the equation is symmetric with respect to the $x\text{-axis}$.

To test the symmetry of the graph with respect to $y\text{-axis}$, replace $x$ by $\text{−}x$ in the equation

$16y^2\text{ = 120(−}x\text{) − 225}$

$16y^2\text{ = −120}x\text{ − 225}$   which is not equivalent to the original equation.

The graph of the equation is not symmetric with respect to the $y\text{-axis.}$

To test the symmetry of the graph with respect to the origin, replace $x$ by $\text{−}x$ and $y$ by $\text{−}y$ in the equation

$16{(-y)}^2\text{ = }120(-x)-225$

$16y^2\text{ = }-120x-225$   which is not equivalent to the original equation.

The graph of the equation is not symmetric with respect to the origin.