My maths teacher told me this problem was impossible without knowledge of implicit differentiation: is she right?
You are given the equation of the circle $\left(x+2\right)^2+\left(y-2\right)^2=16$ , what are the equations of the lines with a gradient of 2 which are tangents of this circle.
Here is a visualisation of the problem 
The red circle is the one I described, and the green and blue lines are the equations I'm looking for.
I think the solution will have something to do with plugging in the value of 2 in some sort of equation for the gradient i.e. $2=\frac{x-x_0}{y-y_0}$ , and then the discriminant the equation you would use to find where they intersect can be solved to find values of $x$ , but I'm not sure how to find these values without using any implicit differentiation -can it be done?
EDIT:
Ok so now I'll try and run this through step by step until i find another problem:
So we can say that there's a line (the diameter) of gradient -1/2 which passes through the centre (-2,2):
$y=mx+c$
$y=-\frac{1}{2}x+c$
$2=1+c$
$y=-\frac{1}{2}x+1$
We can also say that a solution lies on the point $(a,b)$ such that
$\left(a+2\right)^2+\left(b-2\right)^2=16$ and $b=-\frac{1}{2}a+1$
So we can substitute b in the equation of the circle to arrive at
$\left(a+2\right)^2+\left(-\frac{1}{2}a-1\right)^2=16$
which simplifies down to
$\frac{5}{4}a^2+5a-11=0$
and solves to give $\frac{2}{5}\left(-5-4\sqrt{5}\right)$ and $\frac{2}{5}\left(4\sqrt{5}-5\right)$
so these are the possible values for $x$
Now if i substitute 1 of these back into the equation, I'll get two values of $y$ for the one value of $x$ because it's quadratic, but there's only 1 intersection point at each value of $x$ , what do i do at this point?
EDIT 2:
It looks like both the intersection points of the line $x=\frac{2}{5}\left(4\sqrt{5}-5\right)$ correspond to the $y$ values of the intersection points of the tangents, does this mean i only need to plug one of these $x$ values back into the equation?
Visualisation to show you what i mean (the blue line appears to intersect the circle at both the appropriate $y$ values)
Best Answer
Since a tangent to a circle is perpendicular to the circle's radius at that points, we're looking for points $\;(a,b)\;$ on the circle such that the radius through them has a gradient (slope) of $\;-\frac12\;$ , so: as the circle's center is $\;(-2,2)\;$ , the slope from this point to $\;(a,b)\;$ is
$$\frac{b-2}{a+2}=-\frac12\implies -2b+4=a+2$$
but we also have
$$(a+2)^2+(b-2)^2=16\implies16=(-2b+4)^2+(b-2)^2=5(b-2)^2\implies$$
$$b-2=\pm\frac4{\sqrt5}\implies b=2\pm\frac4{\sqrt5}$$
and there you have the $\;y\,-\,$ coordinates of the wanted points. Now find out the $\;x\,-\,$ coordinates and that's all, since once you have the points you can alreayd write down the lines through them as you already have the slope (gradient): $\;2\;$
No calculus at all needed...though, perhaps, it could make things slightly simpler and faster.