Question

* Anyone please give me some information about the following;

• About cone and conic section.(with diagram)
• Describe how are cones, ecllipse, parabola, hyperbola are formed?
(I need full information as much as possible)
Thank you!​

Answer 1

Answer: See explanation below

Explanation:

A conic is a curve obtained as the intersection of the surface of a cone with a plane. The 4 types of conic section are: 1. Hyperbola 2. The Parabola 3. The Ellipse and 4. Circle.

Hyperbola: When the plane cuts the cone at an angle closer to the axis than the side of the cone a hyperbola is formed.

Cones: A cone is a three-dimensional shape that goes in a diagonal directions from a flat base to a point called the apex or vertex. A cone is formed by a set of line segments connecting from the base to the apex.

Parabola: A parabola is the curve formed by the intersection of a plane and a cone, but only when the plane is at the same slope as the side of the cone.

Ellipse: An ellipse is formed by a plane intersecting a cone at an angle to its base. All ellipses have two focal points and a center and a major and minor axis.

Circles: A circle is formed when the plane is parallel to the base of the cone.

*Note: Some of these answers are PLAGIARISED from the internet as I looked these up and copied & pasted some information, so please paraphrase if turning this in.*

Answer 2

Answer: see below

Explanation:

Types of Conics are: Circles, Ellipses, & Hyperbolas.

Here is information about each one:

Circle: Two parabolas facing each other. Distance from the center to the vertices are exactly the same.

(x - h)² + (y - k)² = r²

• (h, k) is the center of the circle
• r is the radius of the circle
• The 4 vertices are: (h, k+r), (h, k-r), (h + r, k), & (h - r, k)

Ellipse: Two parabolas facing each other. Distance from the center to the vertices are not the same.

(x - h)²/a² + (y - k)²/b² = 1

• (h, k) is the center of the ellipse
• a is the horizontal distance from the center to the vertices/co-vertices
• b is the vertical distance from the center to the co-vertices/vertices
• If a > b, then vertices are: (h+a, k), & (h-a, k) and co-vertices are (h, k+b), (h, k-b)
• If b > a, then the vertices and co-vertices are reversed
• Use |a² - b²| = c² to find the distance from the center to the foci.
• If a > b, then Foci = (h+c, k) & (h-c, k). If b > a, then ± from the k-value

Example: (x - 1)²/9 + (y - 2)²/25 = 1

Center (h, k) = (1, 2) a = √9 = 3 b = √25 = 5

Vertices (b > a): (1, 2+5) & (1, 2-5) ==> (1, 7) & (1, -3)

Co-vertices: (1+3, 2) & (1-3, 2) ==> (4, 2) & (-2, 2)

Foci (b > a): c = √(25-9) = 4 --> (1, 2+4) & (1, 2-4) ==> (1, 6) & (1, -2)

Hyperbola: Two parabolas facing AWAY from each other. Distance from the center to the vertices may or may not be the same.

(x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/b² - (x - h)²/a² = 1

↓ ↓

opens left and right opens up and down

• (h, k) is the center of the hyperbola
• If x²-y², then vertices are: (h+a, k), & (h-a, k). There are NO Co-vertices.
• If y²-x², then vertices are (h, k+b), (h, k-b). There are NO Co-vertices.
• Use a² + b² = c² to find the distance from the center to the foci.
• If x²-y², then Foci = (h+c, k) & (h-c, k). If y²-x², then ±c from the k-value
• The slope of the asymptotes are ± b/a and pass through the center (h, k)

Example: (x - 1)²/9 - (y - 2)²/16 = 1

Center (h, k) = (1, 2) a = √9 = 3 b = √16 = 4

Vertices (x²-y²): (1+3, 2) & (1-3, 2) ==> (4, 2) & (-2, 2)

Foci (x²-y²): c = √(9 + 16) = 5 --> (1+5, 2) & (1-5, 2) ==> (6, 2) & (-4, 2)

Asymptotes: m = ± 4/3 through (h, k) = (1, 2) ==> y = ± 4/3(x - 1) + 2