Deriving the Equation of a Circle

What is the radius of a circle whose equation is

(x + 5)² + (y – 3)² = 16?

• 2 units

• 4 units

• 8 units

• 16 units

What is the center of a circle whose equation is

x²+ y² – 12x – 2y + 12 = 0?

• (–12, –2)

• (6, 1)

• (-2, 1)

• (12, 2)

Mrs. Culland is finding the center of a circle whose equation is x² + y² + 6x + 4y – 3 = 0 by completing the square. Her work is shown.

x² + y² + 6x + 4y – 3 = 0

x² + 6x + y² + 4y – 3 = 0

(x² + 6x) + (y² + 4y) = 3

(x² + 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4

Which completes the work correctly?

• (x – 3)² + (y – 2)² = 4², so the center is (3, 2).

• (x + 3)² + (y + 2)² = 4², so the center is (3, 2).

• (x – 3)² + (y – 2)²= 4², so the center is (–3, –2).

• (x + 3)² + (y + 2)² = 4², so the center is (–3, –2).

In the diagram, a circle centered at the origin, a right triangle, and the Pythagorean theorem are used to derive the equation of a circle, x² + y² = r².

If the center of the circle were moved from the origin to the point (h, k) and point P at (x, y) remains on the edge of the circle, which could represent the equation of the new circle?

• (h + x)² + (k + y)² = r²

• (x – h)² + (y – k)²= r²

• (k + x)² + (h + y)² = r²

• (x – k)² + (y – h)² = r²

Which equation represents a circle with a center at (–4, 9) and a diameter of 10 units?

• (x – 9)² + (y + 4)² = 25

• (x + 4)² + (y – 9)² = 25

• (x – 9)² + (y + 4)² = 100

• (x + 4)² + (y – 9)² = 100

What is the equation of the circle, with centre (3,8) and radius 4 units?

• (x - 3)² + (y - 8)² = 4
• (x - 3)² + (y - 8)² = 16
• (x + 3)² + (y + 8)² = 16
• (x + 3)² + (y + 8)² = 4

Find the equation of the circle with center (−8,3) and (2,−5) is on the circle?

Hint: you need to find the radius by plugging in values for four variables or use the distance formula and/or the midpoint formula.

x=

y=

h=

k=

in (x-h)²+(y-k)²=r²

• (x+8)²+(y−3)²=164^.5.

• (x+3)²+(y−8)²=164.

• (x)²+(y)²=164.

• (x+8)²+(y−3)²=164.

What is the circle equation? Hint: Use the equation C=2πr to find the radius?

• (x+12)²+(y+9)²=17π

• (x-9)²+(y-12)²=72.25

• (x+9)²+(y+12)²=72.25

• (x-12)²+(y-9)²=17π

Write the circle equation. Hint: Use the fact that A=πr²

to find the radius.

• (x+2)²+(y-2)²=361

• (x+2)²+(y-2)²=19

• (x-2)²+(y+2)²=19

• (x-2)²+(y+2)²=361

What is the center for the circle with this equation x²+y²+4x+6y-36=0?

• (-2,5)

• (2,7)

• (-2,-3)

• (3,2)