Determine the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1

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Question: Determine the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1.

Options:

Correct Answer: Ellipses with varying axes

Solution:

The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with varying semi-major and semi-minor axes.

Determine the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1.

Determine the family of curves represented by the equation x^2/a^2 + y^2/b^2 = 1.

Correct Answer: Ellipses

Step 1: Identify the equation given, which is x^2/a^2 + y^2/b^2 = 1.
Step 2: Recognize that this equation is in the standard form of an ellipse.
Step 3: Note that 'a' represents the semi-major axis (horizontal stretch) and 'b' represents the semi-minor axis (vertical stretch).
Step 4: Understand that as 'a' and 'b' change, the size and shape of the ellipse will change, creating a family of ellipses.
Step 5: Conclude that the equation represents a family of ellipses with different sizes based on the values of 'a' and 'b'.

Ellipses – The equation represents ellipses, where 'a' and 'b' determine the lengths of the semi-major and semi-minor axes.
Standard Form of Conic Sections – Understanding the standard form of conic sections helps identify the type of curve represented by the equation.