What is the distance between the foci of the ellipse given by the equation 4x^2

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Question: What is the distance between the foci of the ellipse given by the equation 4x^2 + 9y^2 = 36?

Options:

Correct Answer: 6

Solution:

The distance between the foci is 6, calculated using the formula 2c where c = √(a^2 - b^2).

What is the distance between the foci of the ellipse given by the equation 4x^2 + 9y^2 = 36?

What is the distance between the foci of the ellipse given by the equation 4x^2 + 9y^2 = 36?

Step 1: Start with the given equation of the ellipse: 4x^2 + 9y^2 = 36.
Step 2: Rewrite the equation in standard form by dividing everything by 36: (4x^2)/36 + (9y^2)/36 = 1.
Step 3: Simplify the equation: (x^2/9) + (y^2/4) = 1.
Step 4: Identify a^2 and b^2 from the standard form: a^2 = 9 and b^2 = 4.
Step 5: Calculate a and b: a = √9 = 3 and b = √4 = 2.
Step 6: Use the formula to find c: c = √(a^2 - b^2) = √(9 - 4) = √5.
Step 7: Calculate the distance between the foci using the formula 2c: Distance = 2 * √5.
Step 8: Since √5 is approximately 2.236, the distance is approximately 2 * 2.236 = 4.472.
Step 9: However, the exact distance between the foci is 2c = 2 * √5, which is the final answer.

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