Conic Section Lesson: Definition, Types, Formulas, and Examples

Lesson Overview

• What Are Conic Sections?
• Types of Conic Sections
• Circles
• Ellipses
• Parabolas
• Hyperbolas
• Conic Section Formulas
• Parameter of Conic Sections
• Conic Section Examples
• Conic Section Assessment

What Are Conic Sections?

Conic sections are shapes formed when a plane cuts through a cone. These shapes are essential in math and science because they appear in many real-life applications, like the path of planets or the design of bridges.

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What do you know about conic sections?

Types of Conic Sections

1. Circle: Formed when the plane cuts the cone parallel to its base.
2. Ellipse: Formed when the plane cuts through the cone at an angle but does not touch the base.
3. Parabola: Created when the plane cuts through the cone parallel to its side.

Hyperbola: Forms when the plane cuts through both sides of the cone.

Circles

A circle is a set of points that are all the same distance from a central point, called the center.

Fig: Circle

Formula for a Circle:

(x - h)² + (y - k)² = r²
Here:

• (h, k) is the center of the circle.
• r is the radius.

Example :

Find the equation of a circle with a center at (2, 3) and a radius of 5.
Substitute the values into the formula:
(x - 2)² + (y - 3)² = 25

Ellipses

An ellipse looks like a stretched-out circle. It has two fixed points called foci. The sum of the distances from any point on the ellipse to the two foci is always constant.

Fig: Ellipse

Formula for an Ellipse:

(x - h)² / a² + (y - k)² / b² = 1
Here:

• (h, k) is the center.
• a is the distance from the center to the ellipse's horizontal end.
• b is the distance to the vertical end.

Example :

An ellipse has a center at (0, 0), a = 3, and b = 2. The equation is:
x² / 9 + y² / 4 = 1

Parabolas

A parabola is a U-shaped curve. It has a fixed point called the focus and a line called the directrix.

Fig: Parabola

Formula for a Parabola:

Vertical: (x - h)² = 4p(y - k)
Horizontal: (y - k)² = 4p(x - h)
Here:

• (h, k) is the vertex.
• p is the distance from the vertex to the focus.

Example :

A parabola has its vertex at (0, 0) and its focus at (0, 2). The equation is:
x² = 8y

Hyperbolas

Fig: Hyperbola

A hyperbola has two curves that open in opposite directions. It has two foci, and the difference of the distances from any point on the hyperbola to the two foci is always constant.

Formula for a Hyperbola:

Horizontal: (x - h)² / a² - (y - k)² / b² = 1
Vertical: (y - k)² / a² - (x - h)² / b² = 1

Example :

A hyperbola has a center at (0, 0), a = 4, and b = 3. The equation is:
x² / 16 - y² / 9 = 1

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Conic Sections and Circle

Conic Section Formulas

Here is a summary of the key formulas:

Parameter of Conic Sections

Parameters are values like the radius, foci, or distance to the directrix that define a conic section. For example:

• A circle's parameter is its radius.
• An ellipse's parameters are its a, b, and foci.
• A parabola's parameters are its focus and directrix.
• A hyperbola's parameters are a, b, and foci.

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Conic Sections and Circles

Conic Section Examples

1. Circle Example
Find the equation of a circle with a center at (3, -2) and radius 4:
Equation: (x - 3)² + (y + 2)² = 16

Explanation:
The general equation of a circle is:
(x - h)² + (y - k)² = r²
Here, (h, k) is the center of the circle, and r is the radius. In this example:

• The center is (3, -2), so h = 3 and k = -2.
• The radius is 4, so r² = 4² = 16.
  Substitute these values into the formula, giving:
  (x - 3)² + (y + 2)² = 16.

2. Ellipse Example
For an ellipse with center (1, 1), a = 5, and b = 3:
Equation: (x - 1)² / 25 + (y - 1)² / 9 = 1

Explanation:
The general equation of an ellipse is:
(x - h)² / a² + (y - k)² / b² = 1
Here, (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis.

• The center is (1, 1), so h = 1 and k = 1.
• a = 5, so a² = 25.
• b = 3, so b² = 9.
  Substitute these values into the equation, giving:
  (x - 1)² / 25 + (y - 1)² / 9 = 1.

3. Parabola Example
A parabola with vertex (0, 0) and focus at (0, 4):
Equation: x² = 16y

Explanation:
The general equation of a vertical parabola is:
x² = 4py
Here, (h, k) is the vertex, and p is the distance from the vertex to the focus.

• The vertex is at (0, 0), so h = 0 and k = 0.
• The focus is at (0, 4), so p = 4.
  Substitute p = 4 into the formula:
  x² = 4(4)y = 16y.
  This shows the parabola opens upward with a focal length of 4.

4. Hyperbola Example
A hyperbola with center (0, 0), a = 2, and b = 3:
Equation: x² / 4 - y² / 9 = 1

Explanation:
The general equation of a hyperbola is:
x² / a² - y² / b² = 1 (for a horizontal hyperbola)
Here, (h, k) is the center, a is the distance from the center to the vertices along the x-axis, and b is the distance along the y-axis.

• The center is at (0, 0), so h = 0 and k = 0.
• a = 2, so a² = 4.

b = 3, so b² = 9.
Substitute these values into the formula, giving:
x² / 4 - y² / 9 = 1.

Conic Section Assessment

1. Write the equation of a circle with a center at (2, -1) and radius 3.
2. Find the equation of a parabola with vertex (0, 0) and focus at (0, -3).
3. Write the equation of an ellipse with center (0, 0), a = 4, and b = 2.
4. Write the equation of a hyperbola with center (1, -2), a = 3, and b = 5.