<s>[INST] Finding an equation of a circle with a given center and a tangent line. My math homework is finding an equation of the circle. Given that the center is at (-3,-5) and tangent to the line 12x + 5y =4.
I don't know how to solve this since our professor didn't teach this to us. :/
 [/INST] Let $(x+3)^2 + (y+5)^2 = r^2$ be the equation of the circle with center $C = (-3,-5)$ and radius equal to $r$. Rewrite the tangent line's equation as: $y + 5 = \dfrac{29 - 12x}{5}$, substituting this $y$ into the equation of the circle and get:
$(x+3)^2 + \dfrac{(29 -12x)^2}{25} - r^2 = 0 \iff 169x^2 - 546x + 1066 - 25r^2 = 0$. We require that this equation has one and only one real root. This means: $\triangle' = 0 \iff (-273)^2 - 169(1066 - 25r^2) = 0 \iff 4225r^2 - 105625 = 0 \iff r = 5$. So the equation of the circle is: $(x+3)^2 + (y+5)^2 = 25$
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