Plot four points such that they are all exactly 4 units away from the origin.
What shape is formed by connecting these points with four straight lines?
As the number of points we plot and connect gets larger and larger, what shape will be formed?
Consider the graph of the circle shown below:
Complete the statement:
Every point on the circle is exactly β¬ units away from the point (β¬ ,β¬).
Consider the graph of the circle shown below:
State the coordinates of the centre.
State the length of the diameter.
How many points lie on the graph of the following equations:
x^{2} + y^{2} = 0
x^{2} + y^{2} = - 81
Consider the circle with equation x^{2} + y^{2} = 49.
What is the diameter of the circle?
Find the y-values of the points on the circle that have an x-coordinate of 1.
Is the point (- 4, 2) inside, outside or on the circle with equation x^{2} + y^{2} = 21?
How many x-intercepts does the circle \left(x - 7\right)^{2} + \left(y - 1\right)^{2} = 4 have?
The circle \left(x - 3\right)^{2} + \left(y + 1\right)^{2} = 9 is inscribed (fitted exactly) inside a square. What are the coordinates of the vertices of the square?
State the equation of the following circle:
Consider the circle with centre (0, 0) that passes through the point (5, - 10).
Find the radius of the circle.
Find the equation of the circle.
A circle with centre \left(0, 0\right) has an x-intercept at \left(6, 0\right).
Find the equation of the circle.
Find the exact area enclosed by the circle.
For each of the following equations, find:
Consider a circle with centre at \left(5, 4\right) and radius 4.
State the coordinates of the points on the circle that are immediately north and south of the centre.
State the coordinates of the points on the circle that are immediately east and west of the centre.
The equation of a circle is given by x^{2} + \left(y - 3\right)^{2} = 9.
State the coordinates of the centre of this circle.
What is the radius of the circle?
Find the y-intercepts.
Find the x-intercept.
Sketch the graph for the circle.
For each of the following circles:
Sketch the graph.
Find the equation of the circle.
Circle with centre at (3, 3) and a radius of 6 units.
Circle with centre at \left(0, 6\right) and a radius of 2 units.
Write the equation of the circle with its centre at \left( - 5.7 , 0\right) and a radius of 3 units.
If a circle of radius 8 is made to roll along the x-axis, state an equation for the path of the centre of the circle.
Write down the equation of the new circle after x^{2} + y^{2} = 49 is translated:
5 units upwards
5 units downwards
5 units to the right
5 units to the left and 6 units upwards
Consider the circle with centre (8, 6) and radius 8.
Write down the equation of the circle.
Does the circle pass through (0, 0)?
State whether the following are equations of circles:
\left(x + y\right)^{2} = 4
4 x^{2} + 5 y^{2} = 4
y^{2} = 2 + x^{2}
4 x^{2} + 4 y^{2} = 16
x^{2} + y^{2} = 4
y^{2} = 2 - x^{2}
A circle centred at (7, - 8) has an x-intercept at (- 10, 0). Find the exact radius of the circle.
Consider the circle with equation x^{2} + y^{2} = 16.
Find the radius.
Sketch the graph of the circle.
For each of the following equations:
Find the centre of the circle.
Find the radius of the circle.
Graph the circle.
Consider the equation of a circle given by x^{2} + 4 x + y^{2} + 6 y - 3 = 0.
Rewrite the equation of the circle in the form \left(x - h \right)^{2} + \left(y -k\right)^{2} = r^2.
What are the coordinates of the centre of the circle?
What is the radius of the circle?
Consider the equation of a circle given by x^{2} + y^{2} - 2 x - 10 y - 24 = 0.
Rewrite the equation of the circle in the form \left(x - h \right)^{2} + \left(y -k\right)^{2} = r^2.
What are the coordinates of the centre of the circle?
What is the radius of the circle? Express your answer in complexst surd form.
Find the y coordinates of the y-intercepts.
Find the x coordinates of the x-intercepts.
Sketch the graph of the circle.
Consider the equation of a circle given by y^{2} + 2 y + 8 = 12 x - x^{2} + 7.
Rewrite the equation of the circle in the form \left(x - h \right)^{2} + \left(y -k\right)^{2} = r^2.
What are the coordinates of the centre of this circle?
What is the radius of this circle?
Sketch the graph of this circle.
Sketch the graph of the circle x^{2} + y^{2} + 5 x - 3 y = 0.
For the following circles with the given domain and range:
Sketch the graph of the circle.
Find the equation of the circle.
Domain of \left[ - 3 \sqrt{5} , 3 \sqrt{5}\right] and a range of \left[ - 3 \sqrt{5} , 3 \sqrt{5}\right].
Domain of \left[2, 10\right] and a range of \left[ - 4 , 4\right].
A circle has a radius of 1.5 and a centre at \left( - 12 , - 13 \right).
State the domain of the graph in interval notation.
State the range of the graph in interval notation.
Is the point \left( - 13 , - 22 \right) inside or outside of the circle?
Consider the given equation x^{2} + y^{2} - 28 x - 20 y + 100 = 0.
Rearrange the equation into the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}.
State the domain of the graph in interval notation.
State the range of the graph in interval notation.
Consider the equation x^{2} + y^{2} - 32 x + 30 y + 462 = 0.
Rearrange the equation into the form \left(x - h\right)^{2} + \left(y - k\right)^{2} = r^{2}.
State the domain of the graph in interval notation.
State the range of the graph in interval notation.
Determine whether the following points lie inside or outside of the circle.
\left(10, - 15 \right)
\left(16, - 15 \right)
\left(16, -12\right)
\left(25, - 14 \right)
For each of the following equations:
Sketch the graph.
State the domain in interval form.
State the range in interval form.
The top half of a circle has the equation y = \sqrt{ - x^{2} - 24 x + 25} + 11.
Rearrange the equation into the form y = \sqrt{r^{2} - \left(x - h\right)^{2}} + k.
State the domain of the graph in interval notation.
State the range of the graph in interval notation.
Determine whether the following semicircles have the same domain and range:
y = - \sqrt{169 - \left(x + 12\right)^{2}} + 24
y = - \sqrt{169 - \left(x + 12\right)^{2}} + 11
y = \sqrt{169 - \left(x - 12\right)^{2}} - 11
y = \sqrt{169 - \left(x - 11\right)^{2}} - 12
For each of the following semicircles:
State the coordinates of the centre.
A semicircle is described by the function y = \sqrt{64 - x^{2}}.
State the centre of the semicircle.
State the radius of the semicircle.
Hence sketch the graph of y = \sqrt{64 - x^{2}}.
Consider the graph of the function shown below:
State the domain.
State the range.
Find the equation of the following semicircles:
Lower semi-circle centred at \left(0, 0\right) with radius 5.
Right semi-circle centred at \left(0, 0\right) with radius 8.
The top of a semicircle has a domain of \left[ - 10 , 2\right] and a range of \left[ - 2 , 4\right].
Sketch the semicircle.
State the equation for the semicircle in the form y = \pm \sqrt{r^{2} - \left(x - h\right)^{2}} + k.
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 centre 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 centre of the centre circle is in the exact centre of the field. What are the coordinates of the circle's centre?
What is the radius of the centre circle?
State the equation for the circle.
State the domain of the circle in interval notation.
State the range of the circle in interval notation.
The drone can be programmed to only check a rectangular area of the field for whether the ball is within the centre circle or not. What is the smallest area that the drone could check for this task?
Amy needs to use her schoolβs 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 piece of cork.
State the domain and range of the leftmost circle in interval notation.
State the domain and range of the rightmost circle in interval notation.
What is the width of the component that Amy wants to make?
Find the least area of material that could be used to make Amy'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 cm between any cut and the edge of the piece.
If the bottom left corner of the rectangular piece in part (d) is now \left(0, 0\right), state the new equations of the two circles.