How do I find the two tangent points of two lines from (0,0) to an ellipse? We have 2 equations, a general ellipse and it differentiated: 1: A*x*x+B*x*y+C*y*y+D*x+E*y+F=0 is an ellipse if B*B-4*A*C<0. Differentiating, 2: 2*A*x+B*x*dy/dx+B*y+2*C*y*dy/dx+D+E*dy/dx=0. If F<0, ellipse not on (0,0), line from (0,0) tangent to ellipse is: 3: y=dy/dx*x where dy/dx is that of the ellipse. So 3:dy/dx=y/x. Rearranging equation 2: 2: dy/dx=-(2*A*x+B*y+D)/(2*C*y+B*x+E). Combining equation 2 with equation 3: 4: 2*(A*x*x+B*x*y+C*y*y)+D*x+E*y=0. But we haven't satisfied the ellipse equation 1 yet so Solving equations 1 and 4 together, (D*x+E*y)/2+F=0, but that's not a tangent-point solution. How do I get the two solutions of equations 1 and 4? Did I make a mistake, or am I doing it wrong?