Introduction to Conics

Introduction to Conics
The word conic comes from the word cone. 

When we intersect the cone with a plane at varying angles, we are left with the following shapes: parabolas, circles, ellipses, and hyperbolas. 



Conics are also a set of graphs that are derived from the same general equation:

Ax2+Bxy+Cy2+Dx+Ey+F=0\boxed{Ax^2+Bxy+Cy^2+Dx+Ey+F=0}Ax2+Bxy+Cy2+Dx+Ey+F=0​

where A,B,C,D,E,F∈RA, B, C, D, E, F\in\mathbb{R}A,B,C,D,E,F∈R. 
Parabolas: either A=0A=0A=0 or C=0C=0C=0.
Circles: A=CA=CA=C
Ellipses: A,C>0  &  A≠CA,C>0~~\&~~A\neq{}CA,C>0  &  A=C 
Hyperbolas: C<0<AC<0<AC<0<A or A<0<C        ie: A and C are opposite signsA<0<C~~~~~~~~\colorTwo{\footnotesize{\text{ie: A and C are opposite signs}}}A<0<C        ie: A and C are opposite signs

Wize Tip
Completing the square is a critical tool in conics. 

Example: Introduction to Conics
Complete the square in the expression 2x2+12x2x^2+12x2x2+12x.

2x2+12x=2(x2+6x)=2(x2+6x+9)+(−9)(2)=2(x+3)2−18\begin{array}{rcl}
2x^2+12x&=&2(x^2+6x)\\\\
&=&2(x^2+\colorThree{6x}+9)+(-9)(2)\\\\
&=&2(x+3)^2-18
\end{array}2x2+12x​===​2(x2+6x)2(x2+6x+9)+(−9)(2)2(x+3)2−18​

Practice: Introduction to Conics
Indicate whether the following represents a parabola, a circle, an ellipse, or a hyperbola.

a.

2y^2-3x^2=-16

b.

x=-\frac{2}{3}y^2-5

c.

9x^2+12y^2=49

d.

x^2-\frac{1}{8}y=1

e.

-\frac{x^2}{4}+\frac{y^2}{16}=1

f.

-x^2+54y^2=31

I don't know

Practice: Introduction to Conics
Complete the square for the following expression 5y2−25y+145y^2-25y+145y2−25y+14.

A) 5(y−2.5)2−17.255(y-2.5)^2-17.255(y−2.5)2−17.25 

B) 5(y+2.5)2−17.255(y+2.5)^2-17.255(y+2.5)2−17.25 

C) 5(y−2.5)2+45.255(y-2.5)^2+45.255(y−2.5)2+45.25 

D) 5(y+2.5)2+45.255(y+2.5)^2+45.255(y+2.5)2+45.25

I don't know

Practice: Introduction to Conics
Complete the square for the expression x2+12x+4y2+16y=0x^2+12x+4y^2+16y=0x2+12x+4y2+16y=0

A) (x−6)2+4(y−2)2=40(x-6)^2+4(y-2)^2=40(x−6)2+4(y−2)2=40

B) (x+6)2+4(y+2)2=40(x+6)^2+4(y+2)^2=40(x+6)2+4(y+2)2=40 

C) (x−6)2+4(y−2)2=52(x-6)^2+4(y-2)^2=52(x−6)2+4(y−2)2=52 

D) (x+6)2+4(y+2)2=52(x+6)^2+4(y+2)^2=52(x+6)2+4(y+2)2=52 

I don't know

Wize High School Grade 12 Pre-Calculus Textbook > Conics [BC/AB]

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