Hey Jazz. That's an interesting problem in analytic geometry. Here's how I'd write it from that perspective although this may not be what you want:
Given an ellipse and a point on the ellipse (red dot below), place a triangle of height one and base one which is normal to the point with the additional stipulation that the legs of the triangle then be extended down to the arc of the ellipse. Then rotate such figure through an angle
.
Tell you what, just for historical records, I'll post my really messy Mathematica code for doing this and rotating it
. I don't think it's what you want though and Mathematica has built-in functions for doing the transformations but I coded them manually just for fun.
Code:
max = 4;
min = 3/2;
myX = 3;
myY = Sqrt[min^2*(1 - x^2/max^2)] /. x -> myX;
point1 = Graphics[{Red, Point[{myX, myY}]}];
dev = D[Sqrt[min^2*(1 - x^2/max^2)], x] /. x -> myX;
ndev = -dev^(-1);
a = Sqrt[1/(1 + ndev^2)];
b = a*ndev;
x1 = Sqrt[1/(4*(1 + dev^2))];
y1 = x1*dev;
f1[x_] := (((myY + y1) - (myY + b))/((myX + x1) - (myX + a)))*
(x - (myX + a)) + myY + b;
f2[x_] := (((myY + b) - (myY - y1))/((myX + a) - (myX - x1)))*
(x - (myX - x1)) + myY - y1
e1[x_] := Sqrt[min^2*(1 - x^2/max^2)];
xpt = N[x /. First[Solve[f1[x] == e1[x], x]]]
ypt = e1[xpt];
rline = Graphics[Line[{{myX + a, f1[myX + a]},
{xpt, f1[xpt]}}]];
frotate[x_, y_, a_] := {x*Cos[a] - y*Sin[a],
x*Sin[a] + y*Cos[a]};
newrline = Graphics[Line[{frotate[myX + a, f1[myX + a],
alpha], frotate[xpt, f1[xpt], alpha]}]];
xpt2 = N[x /. First[Solve[f2[x] == e1[x], x]]];
ypt = e1[xpt2];
lline = Graphics[Line[{{myX - x1, f2[myX - x1]},
{myX + a, f2[myX + a]}}]];
newlline = Graphics[Line[{frotate[myX - x1, f2[myX - x1],
alpha], frotate[myX + a, f2[myX + a], alpha]}]];
newpoint = Graphics[{Red, Point[frotate[myX, myY, alpha]]}];
c1 = ContourPlot[x^2/max^2 + y^2/min^2 == 1, {x, -5, 5},
{y, -5, 5}, PlotRange -> {{-5, 5}, {-5, 5}},
Axes -> True];
firstplot = Show[{c1, rline, lline, point1}]
alpha = Pi/4;
c2 = ContourPlot[(x*Cos[alpha] + y*Sin[alpha])^2/max^2 +
((-x)*Sin[alpha] + y*Cos[alpha])^2/min^2 == 1,
{x, -5, 5}, {y, -5, 5}, PlotRange -> {{-5, 5}, {-5, 5}},
Axes -> True]
secondplot = Show[{c2, newrline, newlline, newpoint}]
Show[{firstplot, secondplot}]