Question

What are the x- and y-intercepts of the line tangent to the circle (x - 2)^2 + (y - 2)^2 = 25 at the point (5,6)?

Answer 1

Implicit differentiate to get
\[
2 ( x-2) + 2 (y-2) y'=0\\
y'= \frac{2-x}{y-2}\\
\]
Find y' when x=5 and y=6

Answer 2

\[
y'=m = \frac{ 2 -5}{6-2}= -\frac 3 4
\]

Answer 3

\[
y- 6 = -\frac 3 4 ( x-5)
\]

Answer 4

See the circle and the line on the atteched graph.

Answer 5

\[
y- 6 = -\frac 3 4 ( x-5)\\

y = -\frac 3 4 ( x-5) + 6
\]
If x=0 then
\[
y=\frac {39}4
\]
if y=0
\[
x=13
\]