Outcome E15 (adv): solve problems involving the equations and characteristics of circles and ellipses
Originators: Hanna + Erika
Explanation: The equation of a circle in standard form is (x-h)2 + (y-k)2=r2. In transformational form, the equation of a circle is 
where r is the radius, h is the centre x value, k is the centre y value. A circle with an equation of 
will look as such: 
As one can see, The red circle is a basic unit circle with no tranformations [the red circle shown has h=1 and k=1 which are transformations, a circle with no transformations would have a centre at 0,0], and the blue circle is the equation. 3 is the radius, as is seen in the equation as the changed r value. (2, 1) is the centre of the circle, and this can be seen in the equation as the changed (h,k) values.
An ellipse has the equation of: 
and an ellipse with equation 
will look as such: 
The ellipse has a centre at (-2,1) and this is demonstrated in the equation as a change of (h,k) values. The x radius is 3, as can be seen by the changed a value, and the y radius, or major radius, has a value of 5, demonstrated by the changed b value.
Sample Question:
1. Determine the equation in standard form of the circular range of a radar when the radar station is centered at point (5,3) and can see 16 km around it.
2. Determine the equation in standard form of the elliptical shape of an UFO with a center at (0,0) and a major y-axis of 6 and a minor x-axis of 4.
Solution:
Step 1: Determine the values already provided that are within the transformational form of the equations of the circle or ellipse.
1. r = 16, h = 5, k = 3
2. 2a = major y-axis = 6, 2b = minor x-axis = 4, h = -2, k = -1 [The problem above says centre at (0,0). Did you mean (-2, -1)?]
Step 2: Fill in the values into the transformational form of the equation of the circle or ellipse
1. [ 1/16(x - 5) ]2 + [ 1/16(y - 3) ]2 = 1
2. [ 1/3(x + 2) ]2 + [ 1/2(y + 1) ]2 = 1
Step 3: Change to different form if necessary
1. [ 1/16(x - 5) ]2 + [1/16(x - 3) ]2 = 1
(1/16)2 (x - 5)2 + (1/16)2 (y - 3)2 = 1
(1/16)2 ((x- 5)2 + (y - 3)2) = 1
(x - 5)2 + (y - 3)2 = 162
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Comments (1)
Anonymous said
at 7:18 pm on Jun 14, 2007
Nice page Hanna and Erika. I think there are some problems with your sample problem however.
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