Read and use circle equations and graphs

Rule or data

\(\displaystyle d^2=3^2+4^2\)

A point is 3 units horizontally and 4 units vertically from the origin. What is its squared distance from the origin?

Rule or data

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

Which points belong to a circle of radius 5 centred at the origin?

Rule or data

\(\displaystyle x^2+y^2=49\)

Read the radius from this equation

Rule or data

\(\displaystyle C=(2,-1)\)

The centre is (2,−1). Which differences measure a point’s horizontal and vertical displacement from it?

Rule or data

\(\displaystyle (x-h)^2+(y-k)^2=r^2\)

The centre is (h,k) and the radius is r. If h changes from 0 to 2 while k and r stay fixed, what should happen?

Change h while keeping k and r fixed, then change k alone. Finally change r. The axes keep equal scales and a fixed window. C marks the centre

Which description matches the independent changes?

Distance gives the circle equation

Every point on a circle is one radius from its centre. Apply Pythagoras to the horizontal and vertical coordinate differences

1. 1Centre The centre is (h,k). Make each squared bracket zero to read its coordinate
2. 2Radius Take the positive square root of the right-hand side
3. 3Example The centre is (2,−1) and radius is 3. \(\displaystyle (x-2)^2+(y+1)^2=9\)

Rule or data

\(\displaystyle (x-h)^2+(y-k)^2=r^2\)

The radius changes from 2 to 4 without moving the centre. What is the new right-hand side?

Rule or data

\(\displaystyle (x+2)^2+(y-1)^2=9\)

For this circle, what is the greatest possible x-coordinate?

Can a complete circle of positive radius give two y-values for the same x-value?

Rule or data

\(\displaystyle (x-2)^2+(y-1)^2=9\)

To find x-axis intersections, which coordinate must be zero?

Rule or data

\(\displaystyle (x-2)^2+(y-1)^2=9\)

For this circle and x=2, what is the value of (y−1)²?

The circle has centre (2,1) and radius 3. Move the vertical line from x=−2 to x=6. Count its intersections with the fixed circle and compare the squared-coordinate readout. Inspect x=−1, x=2 and x=5 carefully

How does the readout determine the number of real intersections?

Rule or data

\(\displaystyle (x-4)^2+(y+1)^2=9\)

For this different circle, how many intersections are there with x=0?

Rule or data

\(\displaystyle (x+2)^2+(y-3)^2=9\)

For this circle, how many intersections are there with y=0?

Rule or data

\(\displaystyle (x-2)^2+(y-1)^2=25\)

For x=5, enter the smaller y-value on this circle

Rule or data

\(\displaystyle (y-7)^2=16\)

When a translated circle gives (y−7)²=16, which values are correct?

Rule or data

\(\displaystyle (y-1)^2=24\)

Which expression gives both exact y-values?

A diameter has endpoints (−1,−2) and (5,6). What is the centre?

Rule or data

\(\displaystyle r^2=(5-2)^2+(6-2)^2\)

The centre is (2,2) and an endpoint of the diameter is (5,6). Find the radius

Rule or data

\(\displaystyle (x-h)^2+(y-k)^2=25\)

A sprinkler reaches all ground at most 5 m from its centre. What does its circle equation describe?