College Algebra, Chapter 2, 2.4, Section 2.4, Problem 64

a.) Find an equation for the line tangent to the circle $x^2 + y^2 = 25$ at the point $(3, -4)$.

b.) At what other point on the circle will a tangent line be parallel to the tangent line in part (a)?








a.) Since the equation $x^2 + y^2 = 25$ is a circle with center at origin and has radius $5$, we can get the slope of the line that pass through $(0,0)$ and the point of tangency $(3,-4)$

$\displaystyle m = \frac{-4 - 0}{3 - 0} = \frac{-4}{3}$

Then, if we take the negative reciprocal of the slope, we can now determine the slope of the line since both lines are perpendicular to each other, thus, the slope of the line is

$\displaystyle m_T = \frac{3}{4}$

Next, by using Point Slope Form,


$
\begin{equation}
\begin{aligned}

y =& mx + b
&& \text{Model}
\\
\\
y =& \frac{3}{4} x + b
&& \text{Substitute the slope}
\\
\\
-4 =& \frac{3}{4} (3) + b
&& \text{Solve for } b
\\
\\
b =& -4 - \frac{9}{4}
&&
\\
\\
b =& \frac{-25}{4}
&&

\end{aligned}
\end{equation}
$


Thus, the equation of the line is..

$\displaystyle y = \frac{3}{4} x - \frac{25}{4}$


b.) The other point will be the other end point of the diameter that pass through $(3, -4)$ since the circle is symmetric with the origin, the answer is $(-3, 4)$.