What is a closed plane figure with all points at the same distance from the center and satisfying (x-h)^2+(y-k)^2=r^2?

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Study for the Algebra 2 Honors Test. Prepare with flashcards and multiple choice questions, each question comes with hints and detailed explanations. Excel in your exam preparation!

Multiple Choice

What is a closed plane figure with all points at the same distance from the center and satisfying (x-h)^2+(y-k)^2=r^2?

Explanation:
The main idea here is that a circle is the set of all points at a fixed distance from a given center. If the center is (h, k) and the distance from the center to a point (x, y) is r, then by the distance formula this distance is sqrt((x − h)^2 + (y − k)^2). Requiring that distance to equal r gives (x − h)^2 + (y − k)^2 = r^2. This is exactly the equation shown, so the set of all points satisfying it forms a circle with center (h, k) and radius r. A circle is a closed plane figure where every point is the same distance from the center, which matches the description. Why the others don’t fit: an ellipse has an equation with two different radii along perpendicular directions and generally isn’t all points at one fixed distance from the center. A parabola is not closed and does not enclose area in a way where every point is equidistant from a single center. A polygon is made of straight sides and corners, not a smooth curve with constant distance from a center. So the figure described is a circle.

The main idea here is that a circle is the set of all points at a fixed distance from a given center. If the center is (h, k) and the distance from the center to a point (x, y) is r, then by the distance formula this distance is sqrt((x − h)^2 + (y − k)^2). Requiring that distance to equal r gives (x − h)^2 + (y − k)^2 = r^2. This is exactly the equation shown, so the set of all points satisfying it forms a circle with center (h, k) and radius r. A circle is a closed plane figure where every point is the same distance from the center, which matches the description.

Why the others don’t fit: an ellipse has an equation with two different radii along perpendicular directions and generally isn’t all points at one fixed distance from the center. A parabola is not closed and does not enclose area in a way where every point is equidistant from a single center. A polygon is made of straight sides and corners, not a smooth curve with constant distance from a center.

So the figure described is a circle.

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