Originally Posted by

**Soroban** Hello, lindax!

I don't trust your textbook.

**No one** writes the equation of a line like that!

The equation of the circle is: .$\displaystyle x^2+y^2\:=\:9$

Differentiate implicitly: .$\displaystyle 2x + 2yy' \:=\:0\quad\Rightarrow\quad y' \:=\:-\frac{x}{y}$

The line through $\displaystyle (x_o,\,y_o)$ with slope $\displaystyle -\frac{x_o}{y_o}$ .has the equation:

. . . $\displaystyle y - y_o \:=\:-\frac{x_o}{y_o}(x - x_o)\quad\Rightarrow\quad y \:=\:\frac{-x_ox+x_o^2+y_o^2}{y_o}$

Since: .$\displaystyle x_o^2+y_o^2 \:=\:9\:\text{ and }\:y_o \:=\:\pm\sqrt{9-x_o^2} $

. . the equation becomes: .$\displaystyle y \:=\:\frac{-x_ox + 9}{\pm\sqrt{9-x_o^2}} \quad\Rightarrow\quad y \;=\;\pm\frac{9-x_ox}{\sqrt{9-x_o^2}}$