## Implicit Differentiation |
HELP |

Equations involving `x` and `y` define curves in the `xy`-plane. For example,
the graph of the equation `(x^2)/(3^2) + (y^2)/(4^2) = 1` is the set of all
points `(s, t)` in the plane such that substituting `s` for `x` and `t` for `y`
in the equation produces a true statement. In the applet, make `a = 3`, `b =4`,
and `c = 0` to see what this curve looks like. By varying the parameters,
we can see a family of curves, all of which go through the four points `(+-a, 0)`
and `(0, +-b)`. Given a curve, how can we find the slope of a line tangent to
a point on the curve?

**Circle**. Make `a = 5`, `b = 5`, and `c = 0`. Notice that the point `(4,3)` is on the circle. What is the slope of the line tangent to the circle at `(4,3)`?**Ellipse**. Make `a = 4`, `b = 4`, and `c = -0.1` (you can fine-tune `c` by clicking the dot on the slider then using the left and right arrow keys).- What is the slope of the line through `(0, 4)`? (The applet shows the answer to be `4/5`, but use implicit differentiation to determine whether or not this is the exact answer or an approximation.)
- Find the two points where the tangent line is horizontal.

**Parallel(?) Lines**. The graph of the equation `x^2 - xy + 1/4 y^2 = 1` appears to be a pair of parallel lines. See this by setting `a = 1`, `b = 2`, and `c = -1`. Can you prove that these lines are really parallel, or show that they are not?

Enter your answer, in terms of `a`, `b`, `c`, `x`, and `y`:

When you click submit, you'll see the value of your expression in the bottom left corner of the applet. The `x` and `y` values are the `x` and `y` values of the point on the curve. See how the value of your expression compares to the value of the slope computed by the applet.