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 of the circle and r is the radius.

Tags

CCSS.HSG.GPE.A.1

Front

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

Back

To find the center of a circle from its equation in standard form (x-h)² + (y-k)² = r², identify the values of h and k. The center is (h, k).

Tags

CCSS.HSG.GPE.A.1

Front

What does the radius of a circle represent in its equation?

Back

The radius of a circle represents the distance from the center of the circle to any point on the circle. In the equation (x-h)² + (y-k)² = r², r is the radius.

Tags

CCSS.HSG.GPE.A.1

Front

If the equation of a circle is (x+2)² + (y-3)² = 36, what is the radius?

Back

The radius is 6, since r² = 36, and r = √36 = 6.

Tags

CCSS.HSG.GPE.A.1

Front

What is the center of the circle given by the equation (x-5)² + (y+1)² = 25?

Back

The center of the circle is (5, -1).

Tags

CCSS.HSG.GPE.A.1

Front

How do you convert the general form of a circle's equation to standard form?

Back

To convert the general form Ax² + Ay² + Bx + Cy + D = 0 to standard form, complete the square for the x and y terms.

Tags

CCSS.HSG.GPE.A.1

Front

What is the significance of the values h and k in the equation of a circle?

Back

The values h and k represent the x and y coordinates of the center of the circle, respectively.

Tags

CCSS.HSG.GPE.A.1

Front

In the equation (x-3)² + (y-2)² = 16, what does the number 16 represent?

Back

The number 16 represents the square of the radius of the circle. The radius is √16 = 4.

Tags

CCSS.HSG.GPE.A.1

Front

What is the relationship between the radius and the area of a circle?

Back

The area A of a circle is given by the formula A = πr², where r is the radius.

Tags

CCSS.7.G.B.4

Front

How can you determine if a point lies inside, on, or outside a circle?

Back

To determine the position of a point (x₀, y₀) relative to a circle with center (h, k) and radius r, calculate the distance from the point to the center: d = √((x₀-h)² + (y₀-k)²). If d < r, the point is inside; if d = r, it is on the circle; if d > r, it is outside.

Tags

CCSS.HSG.GPE.A.1