Chapter 1.5, Problem 55AYU

55. The Greek Method the Greek method for finding the equation of the tangent line to a circle uses the fact that at any point on a circle the lines containing the center and the tangent line are perpendicular (see Problem 54). Use this method to find an equation of the tangent line to the circle $x^2+y^2=9$ at the point $\left(1,2\sqrt{2}\right)$.

To determine

To find: The tangent line to the circle.

Answer to Problem 55AYU

Solution:

$y=\sqrt{2}\cdot x+4y-9\sqrt{2}=0$

Explanation of Solution

Given:

$x^2+y^2=9$ at the point $\left(1,2\sqrt{2}\right)$

Finding the slope between the center of the circle and the given point

$=\frac{2\sqrt{2}-0}{1-0}=2\sqrt{2}$

Finding the equation of the line that passes through the given point using the slope that perpendicular to the one found above

$2\sqrt{2}=1\cdot \frac{-1}{2\sqrt{2}}+b$

Solving for $b$

$b=\frac{9\sqrt{2}}{4}$

Therefore $y=\frac{-1}{2\sqrt{2}}\cdot x+\frac{9\sqrt{2}}{4}$

That is

$y=\sqrt{2}\cdot x+4y-9\sqrt{2}=0$