Chapter 9.4, Problem 68E

To determine

Determine whether the given statement is true or false.

Answer to Problem 68E

The given statement is False

Explanation of Solution

Given:

The parametric equations

  $x=at+h$ and $y=bt+k$

Where $a+0$ and $b+0$ , represent a circle centered at $(h,k)\text{ when }a=b$

The given statement is false. The parametric equations are given as

  $x=at+h$

  $y=bt+k$

Simplify the above equations to get

  $\begin{array}{l}x–h=at\hfill \\ y–k=bt\hfill \end{array}$

Take $b=a$ in the above equations to get

  $\begin{array}{l}x–h=at\hfill \\ y–k=at\hfill \end{array}$

Now eliminate the parameter t from the two equations to obtain the rectangular equation as

  $\begin{array}{l}y–k=x–h\hfill \\ y=x–h+k\hfill \end{array}$

The last equation represents a straight line in the coordinate plane and not a circle cantered at $\left(h,k\right)$

Therefore the given statement is False.