Chapter 4.1, Problem 50E

To determine

To calculate: To show that for any value of t, the line tangent to the circle at $\left(2\cos t,2\sin t\right)$ is perpendicular to the radius.

Answer to Problem 50E

The slope of radius and the slope of the line of tangent are perpendicular to each other.

Explanation of Solution

Given Information: The coordinates of the circle are $\left(2\cos t,2\sin t\right)$ .

Calculation:

The slope of the line of tangent is,

  $\begin{array}{c}\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\\ =\frac{\frac{d}{dt}\left(2\sin t\right)}{\frac{d}{dt}\left(2\cos t\right)}\\ =\frac{2\cos t}{-2\sin t}\\ =-\cot t\end{array}$

The slope of the radius is,

  $\begin{array}{c}{m}_{\text{radius}}=\frac{y}{x}\\ =\frac{2\sin t}{2\cos t}\\ =\tan t\end{array}$

Clearly, the slope of radius and the slope of the line of tangent are perpendicular to each other.

The slope of radius and the slope of the line of tangent are perpendicular to each other.