Which of the following is the equation of an ellipse with foci at (0, ±c) and ve

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Question: Which of the following is the equation of an ellipse with foci at (0, ±c) and vertices at (0, ±a)?

Options:

Correct Answer: x^2/a^2 + y^2/b^2 = 1

Solution:

The equation of an ellipse with foci at (0, ±c) and vertices at (0, ±a) is y^2/a^2 + x^2/b^2 = 1.

Which of the following is the equation of an ellipse with foci at (0, ±c) and vertices at (0, ±a)?

Which of the following is the equation of an ellipse with foci at (0, ±c) and vertices at (0, ±a)?

Step 1: Understand that an ellipse is a shape that looks like a stretched circle.
Step 2: Know that the foci of the ellipse are points that help define its shape. In this case, the foci are at (0, ±c).
Step 3: Recognize that the vertices are the points where the ellipse is widest. Here, the vertices are at (0, ±a).
Step 4: Remember that for an ellipse centered at the origin with vertical major axis, the general equation is y^2/a^2 + x^2/b^2 = 1.
Step 5: Identify that 'a' is the distance from the center to the vertices, and 'c' is the distance from the center to the foci.
Step 6: Use the relationship c^2 = a^2 - b^2 to find 'b', which is the distance from the center to the ellipse along the x-axis.
Step 7: Conclude that the equation of the ellipse with the given foci and vertices is y^2/a^2 + x^2/b^2 = 1.

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