Question: How would I go about solving the following question?

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

Please provide some hints/clues

Thanks!

Answer: 
1) Find slopes of the tangents to the ellipse at any point on the ellipse. Implicit differentiation helps here, i.e.

$$2 x + 8 y \frac{dy}{dx} = 0$$

2) Find the point(s) on the ellipse whose tangent passes through $(4,6)$. The equation of a line is

$$y-y\_0 = m (x - x\_0)$$

where $m$ is the slope at the point $(x,y)$ on the ellipse and $(x\_0,y\_0)$ is the point $(4,6)$. You end up with an equation like

$$y-6 = \frac{dy}{dx} (x-4)$$

Plug in the derivative for the slope; you will get 2 equations and 2 unknowns (the other equation being the equation for the ellipse). Solution will get you the point(s), and thus the slope(s) and the equation(s) of the lines.

So you can check your results, I get two points: $(4,0)$ (obvious in hindsight) and $(-16/5,6/5)$.
Given a point $(a,\frac 14 \sqrt{16-a^2})$ on the ellipse, you can take the derivative to find the slope of the tangent. You can then use the [point-slope form](http://en.wikipedia.org/wiki/Point-slope_formula#Point.E2.80.93slope_form) for the line and see if it goes through $(4,6)$ If you do this generally, you will get an equation in $a$ that you can solve.