Chapter 6.3, Problem 23E

Solution

To eliminate: The parameter and identify the graph. Use the graph to support your answer.

The general equation is $x^2+y^2=25$ and the graph is shown in figure (1).

Given information:

The equations are $x=5\cos t$ and $y=5\sin t$ .

Calculation:

Add the square of both equations.

  $\begin{array}{c}{x}^2+y^2={\left(5\cos t\right)}^2+{\left(5\sin t\right)}^2\\ x^2+y^2=25{\cos }^{2}t+25{\sin }^{2}t\\ x^2+y^2=25\left({\sin }^{2}t+{\cos }^{2}t\right)\\ x^2+y^2=25\end{array}$

The equation is that of a circle centered at the $\left(0,0\right)$ with radius $5$ from the standard equation ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ . This is supported by the graph obtained from the parametric equations.

  

Figure (1)

Therefore, the general equation is $x^2+y^2=25$ and the graph is shown in figure $\left(1\right)$ .