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).

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- Feb 22nd 2009, 09:05 AMmath123456differentiation question 2
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). - Feb 22nd 2009, 09:33 AMrunning-gag
Hi

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 - Feb 22nd 2009, 09:40 AMTheEmptySet
Here is the idea. Lets suppose that the point of tangency is (a,b)

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

Good luck.

TES