Answer
$y=3$
$y=\frac{2}{3}x-5$
Work Step by Step
$x^2+4y^2=36...(a)\\
2x+8yy'=0\\
y'=-\frac{x}{4y}$
Equation of a tangent line at $(12,3)$
$y-3=m(x-12)\\
y-3=-\frac{x}{4y}(x-12)\\
y-3=\frac{-x^2+12x}{4y}\\
4y^2-12y=-x^2+12x\\
4y^2+x^2=12y+12x$
From $(a)$
$36=12y+12x\\
3=x+y\\
y=3-x...(b)$
Plug in $(b)$ into $(a)$
$x^2+4(3-x)^2=36\\
x^2+4(9+x^2-6x)=36\\
x^2+36+4x^2-24x=36\\
5x^2-24x=0\\
x(5x-24)=0\\
x=0\Longrightarrow y=3\\
x=\frac{24}{5}\Longrightarrow y=-\frac{9}{5}$
First tangent line
$\frac{y-3}{3-3}=\frac{x-0}{12-0}\\
y-3=0\\
y=3$
Second tangent line
$\frac{y-(-\frac{9}{5})}{3-(-\frac{9}{5})}=\frac{x-\frac{24}{5}}{12-\frac{24}{5}}\\
y=\frac{2}{3}x-5$
