I have a question that I'm having difficulty on. I can solve these normally, but I'm having a bit of a challenge dealing with these extra terms:
"Find the equation of the tangent line to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(m,n)$.
Note: your answer should be in terms of $x,y,m,n,a$, and $b$.
On
For an algebraic curve, the simplest is perhaps to consider the associated projective curve: $$\frac{X^2}{a^2}+\frac{Y^2}{b^2}=T^2.$$ The projective line tangent to the curve with homogeneous equation $F(X,Y,T)=0\;$ at a non-singular point $M_0=[X_0:Y_0:T_0]$ has equation: $$XF'_{X}(M_0)X+F'_Y(M_0)Y+F'_T(M_0)T=0$$ The projective point associated to the point $(m,n)$ is $[m:n:1]$ so here you'll you'll get (factoring out $2$: $$\frac{m}{a^2}X+\frac{n}{b^2}Y=T$$ which corresponds to the affine line: $$\frac{m}{a^2}x+\frac{n}{b^2}y=1,\enspace\text{or}\quad mb^2x+na^2y=a^2b^2.$$
By implicitly differentiating the equation for the ellipse, you can find the slope of the tangent line at any point on the ellipse. Then if $(m,n)$ lies on the ellipse, you know the slope of the line through $(m,n)$, so you can find its equation.