12.04 Equations of circles
State the equation for a circle with center \left(h,k\right) and radius r.
Consider the graph of the circle shown.
Copy and complete the following statement:
Every point on the circle is exactly ⬚ units away from the point (⬚ ,⬚).
Consider circle with center C and \triangle ABC with given side lengths. Find the radius of the circle using the Pythagorean Theorem.
Find the distance between Point A and Point B rounded to two decimal places.
A \left(1, 4\right) and B \left(7, 12\right)
A \left(4, 2\right) and B \left( - 8 , - 7 \right)
A \left( - 1 , 9\right) and B \left( - 4 , 1\right)
A \left(-1, - \dfrac{3}{5} \right) and B \left(4, \dfrac{12}{5}\right)
For each of the following circle graphs:
For each of these equations:
Find the center of the circle.
Find the radius of the circle.
Graph the circle.
For each of the these equations for circles:
State the center of the circle.
Calculate the diameter of the circle.
For each of the these descriptions, write the equation of the circle.
A circle with center at (3,\,3) and a radius of 6 units.
A circle with center at \left(0,\,-6\right) and a diameter of 6 units.
A circle with center at \left(\dfrac{17}{2},\,0\right) and a diameter of 2\sqrt{15} units.
A circle with center at \left(\dfrac{1}{2},\,-\dfrac{2}{3}\right) and a radius of \dfrac{5}{4} units.
Using the given points, find the equations for the circles:
Diameter with endpoints at \left(10, 6\right) and \left( - 12 , - 14 \right)
Given: Circle P with center at \left(-3,5\right).
Which equation could represent circle P?
\left(x - 3\right)^2 + \left(y - 5\right)^2 = 41
\left(x - 3\right)^2 + \left(y + 5\right)^2 = 41
\left(x + 3\right)^2 + \left(y - 5\right)^2 = 41
\left(x + 3\right)^2 + \left(y + 5\right)^2 = 41
A circle has a center \left(4,\,0 \right) and goes through the point \left(3,\,-2 \right). Which of these could be the equation of the circle?
Fill in the blanks to complete the equation of the circle.
Circle Q with diameter \overline{GH}: G(-5,\,3) and H(3,\,-1).
Create the equation of this circle.
| The Equation of the Circle | ⬚ | + | ⬚ | = | ⬚ |
|---|
| (x+1)^2 | (x-1)^2 | (y-1)^2 | (y+1)^2 | 20 | 50 |
A circle has a center at ( 3,\, -5) and a radius of 6 units. Create the equation of this circle.
| The Equation of the Circle | ⬚ | ⬚ | ⬚ | = | ⬚ |
|---|
| (x-3) | (x+3) |
| (x-3)^2 | (x+3)^2 |
| (x-5) | (x+5) |
| (x-5)^2 | (x+5)^2 |
| + | - |
| 3^2 | 6^2 |
Find the equation of each circle:
Determine whether the given points are inside, outside, or on the circle with equation x^{2} + y^{2} = 25.
Circle O has a center at (-1,\,3) and a diameter of 10 units.
Which point lies on circle O?
(-6,\,-5)
(2,\,7)
(16,\,4)
(8,\,8)
Which point lies on the circle represented by the equation (x-3)^2 + (y-5)^2 = 5^2?
(0,\,1)
(0,\,4)
(-1,\,5)
(8,\,3)
A wind turbine has 20\text{ m} long blades that are attached to a tower 30\text{ m} high. The distance from the origin to the base of the wind turbine is 25\text{ m}.
Find the standard form of the equation of the circle represented by the path of the blades of the wind turbine.
The circle shown has a radius of 5. Explain how the Pythagorean theorem can be used to derive the equation for this circle.
A soccer match is being televised.
One of the cameras is mounted on a drone which is programmed to zoom in on the ball only when it is inside the center circle.
The drone uses a coordinate system to track the position of the ball, where the origin is at the bottom left corner of the field and each unit corresponds to 1\text{ m}.
The center of the center circle is in the exact center of the field. Find the coordinates of the circle's center.
State the equation for the circle.
State the domain and range of the circle in interval notation.
A circle has a diameter with endpoints of A\left(-1,\,1\right) and B\left(9,\,-7\right).
Determine the equation of the circle.
Prove that the point P\left(8,\,3\right) is outside of the circle.
Coralee needs to use her school’s laser cutter to make a cork component for her design project.
Using technology to sketch the component, she knows that the intersection of two different sized circles inscribed on a piece of cork gives the shape that she wants.
Two such circles have the equations \left(x - 12\right)^{2} + \left(y - 8\right)^{2} = 49 and
\left(x - 20\right)^{2} + \left(y - 8\right)^{2}= 36, where each unit is 1\text{ cm} and \left(0,\,0\right) is the bottom left corner of the given cork board.
State the domain and range of the leftmost circle in interval notation.
State the domain and range of the rightmost circle in interval notation.
Find the width of the component that Coralee wants to make.
Find the least area of material that could be used to make Coralee's shape, considering the following constraints:
The laser cutter will only accept rectangular pieces of material.
The height of the intersection is 10.17\text{ cm}.
There also needs to be a gap of at least 1\text{ cm} between any cut and the edge of the piece.
Three receiving stations are located on a coordinate plane, where each unit is one mile, at the points \left( - 3,\,- 2 \right),\,\left( - 1,\,2\right) and \left(3, 1\right). The epicenter of an earthquake is determined to be 2 miles, 4 miles and 5 miles from these three points, respectively.
Determine the coordinates of the epicenter of the earthquake.
The earthquake can be felt 10 miles away. Determine if the town located at \left(8,\,-7\right) would feel the earthquake.