Equations of Circles

Front

What is the standard form of the equation of a circle?

Back

The standard form of the equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Front

How do you find the center of a circle from its equation?

Back

The center of a circle in the standard form (x - h)² + (y - k)² = r² is given by the coordinates (h, k).

Front

What is the radius of a circle from its equation?

Back

The radius of a circle in the standard form (x - h)² + (y - k)² = r² is the square root of r².

Front

Write the equation of a circle with center (0, 0) and radius 5.

Back

The equation is x² + y² = 25.

Front

What is the distance formula used to find the radius of a circle?

Back

The distance formula is r = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) is the center and (x2, y2) is a point on the circle.

Front

Convert the equation x² + y² - 4x - 6y + 9 = 0 to standard form.

Back

The standard form is (x - 2)² + (y - 3)² = 4.

Front

What does the term 'completing the square' refer to in the context of circles?

Back

Completing the square is a method used to rewrite a quadratic equation in standard form by transforming it into a perfect square trinomial.

Front

If a circle has the equation (x + 3)² + (y - 2)² = 16, what is its center and radius?

Back

Center: (-3, 2), Radius: 4 units.

Front

How do you determine if a point lies on a circle?

Back

Substitute the point's coordinates into the circle's equation. If the equation holds true, the point lies on the circle.

Front

What is the significance of the radius in the equation of a circle?

Back

The radius determines the size of the circle; it is the distance from the center to any point on the circle.