9.6. Degenerate Conics http://www.ck12.org
9.6 Degenerate Conics
Here you will discover what happens when a conic equation can’t be put into graphing form.
The general equation of a conic isAx^2 +Bxy+Cy^2 +Dx+Ey+F=0. This form is so general that it encompasses
all regular lines, singular points and degenerate hyperbolas that look like an X. This is because there are a few
special cases of how a plane can intersect a two sided cone. How are these degenerate shapes formed?
Guidance
Degenerate conic equations simply cannot be written in graphing form. There are three types of degenerate conics:
1.A singular point, which is of the form:(x−ah)^2 +(y−bk)^2 =0. You can think of a singular point as a circle or an
ellipse with an infinitely small radius.
2.A line, which has coefficientsA=B=C=0 in the general equation of a conic. The remaining portion of the
equation isDx+Ey+F=0, which is a line.
3.A degenerate hyperbola, which is of the form: (x−ah)^2 −(y−bk)^2 =0. The result is two intersecting lines that
make an “X” shape. The slopes of the intersecting lines forming the X are±ba. This is becausebgoes with
theyportion of the equation and is the rise, whileagoes with thexportion of the equation and is the run.Example A
Transform the conic equation into standard form and sketch.
0 x^2 + 0 xy+ 0 y^2 + 2 x+ 4 y− 6 = 0
Solution:This is the liney=−^12 x+^32.
Example B
Transform the conic equation into standard form and sketch.
3 x^2 − 12 x+ 4 y^2 − 8 y+ 16 = 0
Solution: 3 x^2 − 12 x+ 4 y^2 − 8 y+ 16 = 0