Chapter 4.2, Problem 2QR

To determine

To calculate: To sketch the curve defined by the equation $4x^2+9y^2=36$ and find two functions $y_1\text{ and }{y}_2$ whose graphs will combine to give the curve.

Answer to Problem 2QR

The graph of the curve is shown above.

The two curves are $y=\sqrt{4-\frac{4}{9}{x}^2}\text{ and }y=-\sqrt{4-\frac{4}{9}{x}^2}$.

Explanation of Solution

Given Information: The equation of the curve is $4x^2+9y^2=36$

Calculation:

The graph of the curve represents an ellipse. The graph of the curve $4x^2+9y^2=36$ is,

  

To find the two functions which combine to form the above curve, simplify the equation of curve,

  $\begin{array}{c}4x^2+9y^2=36\\ \frac{4}{9}{x}^2+y^2=\frac{36}{9}\left(\text{Divide throughout by 9}\right)\\ y^2=4-\frac{4}{9}{x}^2\left(\text{Subtract }\frac{4}{9}{x}^2\text{ from both sides}\right)\\ y=\pm \sqrt{4-\frac{4}{9}{x}^2}\end{array}$

Hence, the two curves are $y=\sqrt{4-\frac{4}{9}{x}^2}\text{ and }y=-\sqrt{4-\frac{4}{9}{x}^2}$