http://www.ck12.org Chapter 9. Conics
3 (x^2 − 4 x)+ 4 (y^2 − 2 y) =− 16
3 (x^2 − 4 x+ 4 )+ 4 (y^2 − 2 y+ 1 ) =− 16 + 12 + 4
3 (x− 2 )^2 + 4 (y− 1 )^2 = 0
(x− 2 )^2
4 +(y− 1 )^2
3 =^0The point (2, 1) is the result of this degenerate conic.
Example C
Transform the conic equation into standard form and sketch.
16 x^2 − 96 x− 9 y^2 + 18 y+ 135 = 0
Solution: 16 x^2 − 96 x− 9 y^2 + 18 y+ 135 = 0
16 (x^2 − 6 x)− 9 (y^2 − 2 y) =− 135
16 (x^2 − 6 x+ 9 )− 9 (y^2 − 2 y+ 1 ) =− 135 + 144 − 9
16 (x− 3 )^2 − 9 (y− 1 )^2 = 0
(x− 3 )^2
9 −(y− 1 )^2
16 =^0This is a degenerate hyperbola.