# Ellipse: Circle and Points

Topics: Circle, Distance, Arc Pages: 12 (2287 words) Published: April 7, 2013
ELLIPSE

9. Given the following equation
9x2 + 4y2 = 36 a) Find the x and y intercepts of the graph of the equation. b) Find the coordinates of the foci.
c) Find the length of the major and minor axes.
d) Sketch the graph of the equation.

Solution
a) We first write the given equation in standard form by dividing both sides of the equation by 36                                         9x2 / 36 + 4y2 / 36 = 1                                         x2 / 4 + y2 / 9 = 1                                         x2 / 22 + y2 / 32 = 1 We now identify the equation obtained with one of the standard equation in the review above and we can say that the given equation is that of an ellipse with a = 3 and b = 2 (NOTE: a >b) . Set y = 0 in the equation obtained and find the x intercepts.                                         x2 / 22  = 1 Solve for x.

x2  = 22                                         x = ± 2 Set x = 0 in the equation obtained and find the y intercepts.                                      y2 / 32 = 1 Solve for y.

y2  = 32                                         y = ± 3 b) We need to find c first.
c2 = a2 - b2 a and b were found in part a).
c2 = 32 - 22                                      c2 = 5 Solve for c.
c = ± (5)1/2 The foci are    F1 (0 , (5)1/2) and  F2 (0 , -(5)1/2)
c) The major axis length  is given by  2 a = 6.
The minor axis length  is given by  2 b = 4.
d) Locate the x and y intercepts, find extra points if needed and sketch.

| How do I find the equation of the ellipse with the given information?

Eccentricity: an index of how circular the ellipse is.

e = c/a

c = the distance from the center of the ellipse to a focal point

a = the distance from the center to a vertex along the major axis

a = 6/2 = 3

e = c/a = √(5) / 3

c/3 = √(5) / 3

c = √(5)

c² = a² - b²

[√(5)]² = 3² - b²

b² = 3² - [√(5)]²

b² = 9 - 5

b² = 4

b = ±2

The general equation for the ellipse will look like this:

x²/b² + y²/a² = 1

The specific equation is:

x²/4 + y²/9 = 1

x²/4 + y²/9 = 1|

Circle

EXAMPLE 1: Write the equation of the circle that passes through the points (2,8), (5,7), and (6,6).

SOLUTION.- The method used in this solution corresponds to the addition-subtraction method used for solution of equations involving two variables. However, the method or combination of methods used depends on the particular problem. No single method is best suited to all problems. First, write the general form of a circle:

For each of the given points, substitute the given values for x and y and rearrange the terms:

To aid in the explanation, we number the three resulting equations:

The first step is to eliminate one of the unknowns and have two equations and two unknowns remaining. The coefficient of D is the same in all three equations and is, therefore, the one most easily eliminated by addition and subtraction. To eliminate D, subtract equation (2) from equation (1):

We now have two equations, (4) and (5), in two unknowns that can be solved simultaneously. Since the coefficient of C is the same in both equations, it is the most easily eliminated variable. To eliminate C, subtract equation (4) from equation (5):

To find the value of C, substitute the value found for B in equation (6) in equation (4) or (5)

Now the values of B and C can be substituted in any one of the original equations to determine the value of D. If the values are substituted in equation (1),

The solution of the system of equations gave values for three independent constants in the general equation

When the constant values are...