SAT circle equations review

Standard equation of a circle?

The standard form is $\displaystyle (x - h)^2 + (y - k)^2 = r^2$ where (h, k) is the center and r is the radius.

Identify center and radius: $\displaystyle (x - 3)^2 + (y + 2)^2 = 16$

Center: (3, -2) Radius: 4 (since $\displaystyle r^2 = 16$)

True or False: Circle equations can be written in multiple forms.

True. They can be expressed in standard form or converted to general form.

Convert $\displaystyle (x + 1)^2 + (y - 4)^2 = 9$ to general form.

Expand: $\displaystyle x^2 + 2x + 1 + y^2 - 8y + 16 = 9$ Combine: $\displaystyle x^2 + y^2 + 2x - 8y + 8 = 0$

Fill in the blank: The circle with center (0,0) and radius 5 is ____.

$\displaystyle (x)^2 + (y)^2 = 25$

What happens if the radius is zero?

The circle becomes a point at the center (h, k).

Identify the radius from: $\displaystyle (x - 7)^2 + (y + 1)^2 = 36$

Radius: 6 (since $\displaystyle r^2 = 36$)

Standard vs General Form: circle equations.

Standard: $\displaystyle (x - h)^2 + (y - k)^2 = r^2$ General: $\displaystyle Ax^2 + Ay^2 + Bx + Cy + D = 0$

Given center (2, -3), radius 5, write the equation.

$\displaystyle (x - 2)^2 + (y + 3)^2 = 25$

True or False: The center of the circle affects its size.

False. The size is determined by the radius only.

What form is: $\displaystyle x^2 + y^2 + 4x - 6y + 9 = 0$?

General form. It can be converted to standard form.

Effect of changing (h, k) on circle equation?

It shifts the circle's center without changing its radius.

Expand: $\displaystyle (x - 4)^2 + (y + 5)^2 = 49$.

$\displaystyle x^2 - 8x + 16 + y^2 + 10y + 25 = 49$ Combine: $\displaystyle x^2 + y^2 - 8x + 10y - 8 = 0$

Identify center: $\displaystyle x^2 + y^2 - 6x + 8y + 9 = 0$

Center: (3, -4) (Complete the square to find center.)

What is the radius if: $\displaystyle (x + 2)^2 + (y - 3)^2 = 1$?

Radius: 1 (since $\displaystyle r^2 = 1$)

Write the circle equation in standard form.

To write the equation in standard form, rearrange: 1. Start with general form: $\displaystyle x^2 + y^2 + Dx + Ey + F = 0$. 2. Complete the square for $\displaystyle x$ and $\displaystyle y$. Example: Convert $\displaystyle x^2 + y^2 - 8x - 6y + 12 = 0$.

A circular park has a radius of 30 feet. What is its area?

Area = \displaystyle C0 r^2 = \displaystyle C0 (30)^2 = 900\displaystyle C0 square feet.

True or False: The circumference of a circle is always greater than its diameter.

True. Circumference = \displaystyle C0 d; it's always more than the diameter.

Fill in the blank: The formula for circumference is ____.

Circumference = \displaystyle C = C0 d or \displaystyle C = 2C0 r.

How far does a car travel on a circular track of radius 50 meters?

Circumference = \displaystyle C = 2C0(50) C4 A6 C6 B1 BF = 100\displaystyle C0 meters.

Compare the areas of circles with radii 5 and 10.

Area of radius 5: \displaystyle 25C0; Radius 10: \displaystyle 100C0. The larger circle's area is 4 times bigger.

What is the diameter of a circle with an area of 200\displaystyle C0?

Area = 200\displaystyle C0; $\displaystyle r^2 = 200$; $\displaystyle r = 10 ext{ (diameter = 20)}$.

If a circular swimming pool has a radius of 12 feet, what is its circumference?

Circumference = \displaystyle C = 2C0(12) = 24C0 feet.

True or False: A circle can have a radius of zero.

True. A radius of zero means a point circle.

Calculate the area of a circle that has a diameter of 16 inches.

Radius = 8 inches; Area = \displaystyle 64C0 square inches.

What distance does a bicycle cover after 3 laps on a circular track with radius 15 yards?

Circumference = \displaystyle 30C0 yards; Total distance = \displaystyle 3 imes 30C0 = 90C0 yards.

If the radius doubles, how is the area affected?

Area quadruples. If $\displaystyle r$ becomes $\displaystyle 2r$, then area = $\displaystyle 4 ext{(old area)}$.

Determine the radius of a circle with a circumference of 31.4 meters.

Using \displaystyle C = 2C0 r: $\displaystyle r = 5$ meters.

True or False: The area of a circle increases linearly with radius.

False. Area increases with the square of the radius.

A pizza has a radius of 10 inches. What is the area?

Area = \displaystyle 100C0 square inches.

If a circular garden has a diameter of 18 feet, what's the radius?

Radius = $\displaystyle \frac{18}{2} = 9$ feet.

What shape is formed by connecting points equidistant from a center?

A circle.

Calculate the circumference of a wheel with a radius of 1.5 feet.

Circumference = \displaystyle 3C0 feet.

During a fair, a circular ride has a radius of 20 feet. What is the area?

Area = \displaystyle 400C0 square feet.

If a circular pizza is cut into 8 equal slices, what angle does each slice have?

Each slice: $\displaystyle 360^ ext{°} / 8 = 45^ ext{°}$.

If two circles have the same area, what can be said about their radii?

Their radii are equal.