Could you please show me step by step how to solve this problem?
Suppose that the equation of an ellipse is 36X^2 + 81y^2 = 2916
Find the major and minor radii. Use these numbers and the realtionship in problem 11 to calculate c, the focal readius for this elipse.
number 11 stated exlapin why b^2 + c^2 = a^2. Write this relationship on your index card.
THANK YOU!
l was not even born yet in jun 9 2009 3 month later l was born
Sure! I'd be happy to help you solve this problem step by step.
Step 1: Write down the equation of the ellipse: 36X^2 + 81y^2 = 2916
Step 2: Compare the given equation to the standard form of the equation of an ellipse:
(x^2/a^2) + (y^2/b^2) = 1
From the given equation, we can see that a^2 = 36 (taking square root on both sides gives a = 6), and b^2 = 81 (taking square root on both sides gives b = 9).
Step 3: Identify the major and minor radii:
The major radius (2a) of an ellipse is twice the length of the major axis. In this case, the major radius is 2a = 2 * 6 = 12.
The minor radius (2b) of an ellipse is twice the length of the minor axis. In this case, the minor radius is 2b = 2 * 9 = 18.
Step 4: Use the relationship b^2 + c^2 = a^2 to find c, the focal radius for this ellipse:
Given that a^2 = 36, and b^2 = 81, we can substitute these values into the equation.
b^2 + c^2 = a^2
81 + c^2 = 36
Rearrange the equation to isolate c^2:
c^2 = 36 - 81
c^2 = -45
Note that since c^2 is a negative value, there is no real solution for c in this case.
Step 5: Write down the relationship b^2 + c^2 = a^2 on your index card for reference.
So, the major radius is 12 units, the minor radius is 18 units, and there is no real solution for c (the focal radius) in this ellipse.
I hope this helps! Let me know if you have any further questions.
The canonical form for an ellipse is
(x/a)2 + (y/b)2 = 1
where a and b are the major and minor radii respectively (note: major>minor).
If you rewrite
36X2 + 81y2 = 2916
as
(6x)2 + (9y)2 = 542
you should have no problem finding a and b.
See also:
http://en.wikipedia.org/wiki/Ellipse