Chapter 9 Quadratic Relations and Conic Sections - 9.3 Graph and Write Equations of Circles - 9.3 Exercises - Skill Practice - Page 630: 57

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From geometry, a line tangent to a circle is perpendicular to the radius at the point of tangency. The radius with endpoint $(-5,9)$ has slope $m=\frac{0-9}{0+5}=-\frac{9}{5}$, so the slope of the tangent line at $(-5, 9)$ is the negative reciprocal of $\frac{5}{9}$. An equation of the tangent line is as follows: $$y-9=\frac{5}{9}(x+5)\\y=\frac{5}{9}x+\frac{106}{9}$$