(*******
Find the equation of the tangent line and the normal line to the graph of 3x^2 + 4y^2=7 at point (1, 1)
*****)

f =3x^2 + 4y^2;
n=grad[x_,y_] = {D[f,x],D[f,y]}

p = {1,1};
n = grad[1,1]

(***
Normal line:
**)
n.{x,y} == n.p
r[t_] = p + t n
curve = ContourPlot[3x^2 + 4y^2,{x,-2,2},{y,-2,2},PlotPoints->50, Contours->{7}, ContourShading->False];

nline=ParametricPlot[{1+6t,1+8t},{t,-1,1}];
tline=ParametricPlot[{1+8t,1-6t},{t,-1,1}];

Show[curve,nline,tline,AxesLabel->{x,y}]