Page 102 - Calc for the Clueless

This is weird. No matter what the coefficient on the right side, factor the whole coefficient out,
even if there is a fraction in the parentheses.

Sketch v(-2,-1/3), shape 2, . 4c = 3, c = 3/4. F(-2,-1/3,-3/4). Directrix y = -1/3 + 3/4.

We will now look at the ellipse. Algebraically, the ellipse is defined as PF 1 + PF 2 = 2a, where 2a > 2c, the
distance between F 1 and F 2. In words, given two points, F 1 and F 2, two foci. If we find all points P, such that if
we go from F 1 to P and then from P to F 2, add those two distances together, and we will always get the same
number, 2a, where a will be determined later; we will get an ellipse.

I know you would desperately like to know how to draw an ellipse. This is how. Take a nonelastic string.
Attach both ends with thumbtacks to the table. Take the point of a pencil and stretch the string as far as it will
go. Go 360 degrees. You will trace out an ellipse.

Some of you have seen the equation for an ellipse, but few of you have seen the derivation. It is an excellent
algebraic exercise for you to try. You will see there is a lot that goes into a rather simple equation.

Combine like terms; isolate the
radical.

Reverse sides; take out common
factors.

2
2
2
Divide on both sides by a (a - c ).