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.
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Sure! Let's break down the steps to solve this problem:
Step 1: Recognize the equation of the ellipse.
The given equation is 36X^2 + 81y^2 = 2916. The general equation of an ellipse is (x^2 / a^2) + (y^2 / b^2) = 1, where a is the major radius and b is the minor radius.
Step 2: Rewrite the given equation in the general form.
To do this, we divide each term by 2916 to make the right side of the equation equal to 1:
(36X^2 / 2916) + (81y^2 / 2916) = 1
Simplifying, we get:
X^2 / 81 + y^2 / 36 = 1
Comparing this with the general form, we can see that a^2 = 81 and b^2 = 36.
So, the major radius (a) is 9 and the minor radius (b) is 6.
Step 3: Calculate c, the focal radius.
To calculate c, we use the relationship b^2 + c^2 = a^2.
Substituting the values:
6^2 + c^2 = 9^2
36 + c^2 = 81
Subtracting 36 from both sides, we have:
c^2 = 81 - 36
c^2 = 45
Taking the square root of both sides, we find:
c = √45
Therefore, the focal radius (c) is √45.
Remember to write the relationship b^2 + c^2 = a^2 on your index card.
I hope this helps! Let me know if you have any further questions.