Precalculus A 2016: Rectangular and Parametric Forms of Conic Sections
10-6
conic
Ax2 Bxy Cy2 Dx Ey F 0
circle
AC
parabola
Either A or C is zero.
ellipse
A and C have the same sign and A C.
hyperbola
A and C have opposite signs.
Identify the conic section represented by each equation. Then
write the equation in standard form and graph the equation.
1. x 2 4y 4 0
2. x 2 y 2 6x 6y 18 0
3. 4x 2 y 2 8x 6y 9
4. 9x 2 5y 2 18x 36
General Form: Ax 2 Bxy C y 2 Dx Ey F 0
Conic Section
Standard Form
of Equation
Variation of General
Form of Conic Equations
circle
(x h)2 (y k)2 r2
AC
parabola
(y k)2 4p(x h) or
(x h)2 4p(y k)
Either A or C is zero.
ellipse
hyperbola
(x h)2
(y k)2
1 or
2
a
b2
(y k)2
(x h)2
1
2
a
b2
(x h)2
(y k)2
1 or
a2
b2
(y k)2
(x h)2
1
a2
b2
xy k
A and C have the
same sign and A C.
A and C have opposite signs.
ACDE0
5. Identify the conic section represented by 4y2 10x 16y x 2 5. Write the
equation in standard form.
6. In the general equation of a conic, A C 2, B 0, D 8, E 12, and F 6.
Write the equation in standard form.