Question

Find the equations of both of the tangent lines to the unit
circle x^2 + y^2 = 1 that pass through the point (5, 5).

Answer 1

step 1 : find slope of tangent lines by getting dy/dx using implicit differentiation
\[2x +2y \frac{dy}{dx} = 0\]
\[\frac{dy}{dx} = -\frac{x}{y}\]

step 2: equate the slope of tangent (dy/dx) to slope obtained by point (5,5) and point on circle (x1,y1)
- this will allow you to solve for the 2 points on circle the lines go through
\[-\frac{x_1}{y_1} = \frac{y_1 -5}{x_1 -5}\]
from unit circle , you know that
\[y_1 = \sqrt{1-x_1^{2}}\]
by substitution you can solve for (x1,y1) ...there are 2 solutions
\[\rightarrow (\frac{4}{5}, -\frac{3}{5})\]
\[\rightarrow (-\frac{3}{5}, \frac{4}{5})\]

step 3: determine slope of each line from points found above, get line equations using given point (5,5)
\[y-5 = m(x-5)\]
\[m_1 = -\frac{x_1}{y_1} = -\frac{4/5}{-3/5} = \frac{4}{3}\]
\[m_2 = -\frac{x_2}{y_2} = -\frac{-3/5}{4/5} = \frac{3}{4}\]
which yields the following 2 tangent lines:
\[y = \frac{4}{3}x -\frac{5}{3}\]
\[y = \frac{3}{4}x +\frac{5}{4}\]
here is graph of solution:
http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%3D1+%2C+y%3D%283%2F4%29x%2B5%2F4+%2C+y%3D%284%2F3%29x-5%2F3