Introduction to Conic Sections

Question 1

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.

Answer 1

Conic Section or Conic

Question 2

If the plane passes through the vertex of the cone. It may be a point, line, or two intersecting lines.

Answer 2

Degenerate Conic.

Question 3

When the cutting plane is perpendicular to the axis of the cone.

Answer 3

Circle

Question 4

When the cutting plane is parallel to one side of the cone.

Answer 4

Parabola

Question 5

When the cutting plane is oblique to the axis of the cone.

Answer 5

Ellipse

Question 6

When the cutting plane is parallel to the axis of the cone

Answer 6

Hyperbola

Question 7

General equation of a conic

Answer 7

Ax^2+Bxy+Cy^2+Dx+Ey+F=0

Question 8

Classifying Conic Sections: Circle

Answer 8

A = C

Question 9

Classifying Conic Sections: Ellipse

Answer 9

A and C have the same sign

Question 10

Classifying Conic Sections: Parabola

Answer 10

A=0 or C=0

Question 11

Classifying Conic Sections: Hyperbola

Answer 11

A and C have opposite signs

Question 12

Classifying Conic Sections:

x^2+3y^2−4x+6y−13=0

Answer 12

Ellipse

Question 13

Classifying Conic Sections:

3x^2+3y^2+6x+6y−4=0

Answer 13

Circle

Question 14

Classifying Conic Sections:

4y^2−2x+8y+10=0

Answer 14

Parabola

Question 15

Classifying Conic Sections:

3x^2+5x−y+2=0

Answer 15

Parabola

Question 16

Classifying Conic Sections:

5x^2−4y^2−10x+16y−25=0

Answer 16

Hyperbola

Question 17

Classifying Conic Sections:

−x^2−y^2+3x−5y+20=0

Answer 17

Circle

Question 18

Standard Equation of a Circle

Answer 18

(𝑥 − ℎ)^2 + (𝑦 − 𝑘)^2 = 𝑟^2

Question 19

Standard Equation of an Ellipse

Answer 19

((x-h)^2/a^2) + ((y-k)^2/b^2) = 1

Question 20

Standard Equation of a Hyperbola

Answer 20

((x-h)^2/a^2) - ((y-k)^2/b^2) = 1

Question 21

Standard Equation of a Parabola

Answer 21

(𝑥 − ℎ)^2 = ±4𝑝(𝑦 − 𝑘)

Question 22

Complete the Square:

𝑥^2 + 8𝑥 = 0

Answer 22

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

Question 23

Complete the Square:

𝑦^2 − 6𝑦 = 0

Answer 23

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

Question 24

Completing the Square Equation

Answer 24

(b/2)^2

Question 25

Transform the general equation to standard equation:

x^2+y^2−2x−4y+1=0

Answer 25

• 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

Question 26

Transform the general equation to standard equation:

9x^2+4y^2−36x−108=0

Answer 26

• 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