Find the equations of the lines that are tangents to the elipse
x^2 + 4y^2 = 16 and that also passes through the point (4,6).
First the line is tangent to the ellipse and passes through the point (4,6).
If there is another one its equation is
It is tangent to the ellipse. Therefore there is only one contact point between the ellipse and the line. In other words the system
has one single solution
It means that the equation or has one solution. Its discriminant is thus equal to 0.
The line passes through the point (4,6). Therefore or .
Then giving and
Plugging this point in we get our 1st equation
Now lets take the derivative of the equation(implicitly)
So now we know the slope is
We also know that the tangent line passes through (4,6) so we get the slope is
and we know that the slopes have to be equal.... so setting them equal yields our 2nd equation.
This is a system of non-linear equations...
Solveing yeilds the solution
So plugging into the equation for the derivative we get
So we end up with