Introduction to Conic Sections Flashcards
If a plane is made to cut a double right circular cone. If the plane does not pass through the vertex of the cone, then the conic sections formed are: a circle, an ellipse, a parabola and a hyperbola.
Conic Section or Conic
If the plane passes through the vertex of the cone. It may be a point, line, or two intersecting lines.
Degenerate Conic.
When the cutting plane is perpendicular to the axis of the cone.
Circle
When the cutting plane is parallel to one side of the cone.
Parabola
When the cutting plane is oblique to the axis of the cone.
Ellipse
When the cutting plane is parallel to the axis of the cone
Hyperbola
General equation of a conic
Ax^2+Bxy+Cy^2+Dx+Ey+F=0
Classifying Conic Sections: Circle
A = C
Classifying Conic Sections: Ellipse
A and C have the same sign
Classifying Conic Sections: Parabola
A=0 or C=0
Classifying Conic Sections: Hyperbola
A and C have opposite signs
Classifying Conic Sections:
x^2+3y^2−4x+6y−13=0
Ellipse
Classifying Conic Sections:
3x^2+3y^2+6x+6y−4=0
Circle
Classifying Conic Sections:
4y^2−2x+8y+10=0
Parabola
Classifying Conic Sections:
3x^2+5x−y+2=0
Parabola
Classifying Conic Sections:
5x^2−4y^2−10x+16y−25=0
Hyperbola
Classifying Conic Sections:
−x^2−y^2+3x−5y+20=0
Circle
Standard Equation of a Circle
(𝑥 − ℎ)^2 + (𝑦 − 𝑘)^2 = 𝑟^2
Standard Equation of an Ellipse
((x-h)^2/a^2) + ((y-k)^2/b^2) = 1
Standard Equation of a Hyperbola
((x-h)^2/a^2) - ((y-k)^2/b^2) = 1
Standard Equation of a Parabola
(𝑥 − ℎ)^2 = ±4𝑝(𝑦 − 𝑘)
Complete the Square:
𝑥^2 + 8𝑥 = 0
𝑥^2 + 8𝑥 + (8/2)^2= 0 + (8/2)^2
𝑥^2 + 8𝑥 + (4)^2 = (4)^2
𝑥^2 + 8𝑥 + 16 = 16
(𝑥 + 4)^2 = 16
Complete the Square:
𝑦^2 − 6𝑦 = 0
𝑦^2 − 6𝑦 + (6/2)^2= 0 + (6/2)^2
𝑦^2 − 6𝑦 + (3)^2 = (3)^2
𝑦^2 − 6𝑦 + 9 = 9
(𝑦-3)^2 = 9
Completing the Square Equation
(b/2)^2
Transform the general equation to standard equation:
x^2+y^2−2x−4y+1=0
- A=1 and C=1
- since A=C, the equation is a circle
- the standard equation of a circle is of the form
(x−ℎ)^2+(y−k)^2=r^2
(x^2−2x)+(y^2−4y)=−1
(x^2−2x+(2/2)^2)+(y^2−4y+(4/2)^2)=−1+(2/2)^2+(4/2)^2
(x^2−2x+1)+(y^2−4y+4)=−1+1+4
(x−1)^2+(y−2)^2=4
Transform the general equation to standard equation:
9x^2+4y^2−36x−108=0
- A=9 and C=4
- since both A and C have the same sign, the equation is an ellipse
- the standard equation of an ellipse is of the form
((x−ℎ)^2/a^2)+((y−k)^2/b^2)=1.
(9x^2−36x)+4y^2=108
9(x^2−4x)+4y^2=108
9(x^2−4x+(4/2)^2)+4y^2=108+9(4/2)^2
9(x^2−4x+4)+4y^2=108+36
9(x−2)^2+4y^2=144
9(x−2)^2/144+4y^2/144=144/144
(x−2)^2/16+y^2/36=1