1. implicit differentiation question

find the equations of the lines that are tangent to the eclipse x^2 + 4y^2 =16 and that also pass through the point (4,6)

i implicitly differentiated it and i got -x/4y

then i plugged the point (4,6) to find the slope and got an equation that is 6y + x -40 = 0 but according to the text the answers are x-4=0 and 2x-3y + 10 = 0

what did *I do wrong and how do i solve this problem using implicit differentiation.

edit: typo

2. Re: implicit differentiation question

The derivative gives the slope of the the curve at any point *on the curve*. The point (4,6) is not on the curve.

For any point $(x_0,y_0)$ on the ellipse, you can calculate the tangent line as $y-y_0 = -\frac{x_0}{4y_0}(x-x_0)$. Plugging in $(x,y)=(4,6)$ gives you $y_0$ as a function of $x_0$. You can then plug this into $x_0^2 + 4y_0^2 =16$ to get a quadratic equation in $x_0$. The two solutions should give you the two tangent lines.

- Hollywood