9.1. General Form of a Conic http://www.ck12.org
Solution: First write out the equation with space so that there is room for the terms to be added to both sides. Since
this is an equation, it is appropriate to add the values to both sides instead of adding and subtracting the same value
simultaneously. As you rewrite with spaces, factor out any coefficient of thex^2 ory^2 terms since your algorithm for
completing the square only works when this coefficient is one.
x^2 + 6 x+ + 2 (y^2 + 8 y+ ) = 0
Next complete the square by adding a nine and what looks like a 16 on the left (it is actually a 32 since it is inside
the parentheses).
x^2 + 6 x+ 9 + 2 (y^2 + 8 y+ 16 ) = 9 + 32
Factor.
(x+ 3 )^2 + 2 (y+ 4 )^2 = 41
Concept Problem Revisited
The one essential skill that you need for conics is completing the square. If you can do problems like Example C
then you will be able to graph every type of conic.
Vocabulary
Completing the squareis a procedure that enables you to combine squared and linear terms of the same variable
into a perfect square of a binomial.
Conicsare a family of graphs (not functions) that come from the same general equation. This family is the
intersection of a two sided cone and a plane in three dimensional space.
Guided Practice
- Identify the type of conic in each of the following relations.
a. 3x^2 = 3 y^2 + 18
b.y= 4 (x− 3 )^2 + 2
c.x^2 +y^2 = 4
d.y^2 + 2 y+x^2 − 6 x= 12
e. x 62 + 12 y^2 = 1
f. x^2 −y^2 + 4 = 0- Complete the square in the following expression.
6 y^2 − 36 y+ 4 - Complete the square for bothxandyin the following equation.
− 3 x^2 − 24 x+ 4 y^2 − 32 y= 8
Answers: - a. The relation is a hyperbola because when you move the 3y^2 to the left hand side of the equation, it becomes
negative and then the coefficients ofx^2 andy^2 have opposite signs.
b. Parabola
c. Circle
d. Circle
e. Ellipse