Page 75 - Calculus for the Clueless, Calc II

In the case of the parabola, what has changed is the vertex. Instead of being at the point (0,0), the vertex is at
the point (-2,3). The shape is the same. The value for 4c is still 7. So c = 7/4. The focus now becomes (-2 -
7/4,3), 7/4 to the left of the vertex (-7/4 from the x coordinate). The directrix is x = -2 + 7/4.

Example 5—

Sketch the parabola 2X + 8X + 6y + 10 =0:
2
Original

Divide through by the coefficient of the squared variable.

On one side, get all the terms that have the squared letter; everything else to the other side.

Complete the square; add to both
sides.

Factor and simplify.

Weird thing. No matter what the coefficient on the right side is, 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, you get 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 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 anonelastic string.
Attach both ends with thumbtacks to the table. Take the pencil point and stretch the string as far as it will go.
Go 360º. You will trace out an ellipse.