Circles
Identify the centre and radius of a circle from standard form; complete the square to convert general form to standard form; find the equation of a circle given its centre and a point on the circumference, and state its intercepts.
Worked examples
Identifying centre, radius, domain and range from standard form
Straightforward
Problem
A circle has equation . State the centre and radius, then write the domain and range.
1
Compare the equation with the standard form to read off , and .
2
Find the radius by taking the positive square root of .
3
State the centre and radius.
4
Find the domain. The x-values extend units either side of the centre's x-coordinate.
5
Find the range. The y-values extend units either side of the centre's y-coordinate.
Answer
Centre , radius , domain , range
Completing the square to find standard form
Moderate
Problem
Rewrite in standard form and state the centre and radius.
1
Group the terms and terms together and move the constant to the right.
2
Complete the square for : take half the coefficient of , square it, and add it to both sides. Half of is , and .
3
Complete the square for : half of is , and . Add to both sides.
4
Write each group as a perfect square and simplify the right-hand side.
5
Read off the centre and radius from the standard form .
Answer
; centre , radius
Finding the equation of a circle from its centre and a point, then finding intercepts
Challenging
Problem
A circle has centre and passes through the point . Find the equation of the circle in standard form, then state all x- and y-intercepts.
1
Find using the distance formula from the centre to the point .
2
Write the equation of the circle in standard form with centre .
3
Find the x-intercepts by substituting and solving for .
4
Find the y-intercepts by substituting and solving for .
5
State all intercepts.
Answer
Equation: ; x-intercepts: and ; y-intercepts: and
Practise
Q1·Straightforward
A circle has equation . What is the radius of the circle?
Explanation
Comparing with the standard form gives , so .
Q2·Straightforward
The circle has centre . What is the value of ?
Explanation
Writing as shows that . So the centre is and .
Q3·Straightforward
The circle has domain . What is the value of ?
Explanation
The centre is and , so . The x-values range from to . Therefore the domain is and .
Q4·Straightforward
Which statement correctly describes the circle ?
Explanation
is the same as . Comparing with gives centre and radius .
Q5·Moderate
Rewrite in standard form. What is the radius of the circle?
Explanation
Group and complete the square:\n
\n
\n
\nSo and .
Q6·Moderate
Rewrite in standard form and find the x-coordinate of the centre.
Explanation
Complete the square:\n
\n
\nThe centre is , so the x-coordinate is .
Q7·Moderate
The circle is rewritten in standard form. What is the radius?
Explanation
Complete the square:\n
\n
\nSo and .
Q8·Moderate
Rewrite in standard form and find the y-coordinate of the centre.
Explanation
Complete the square:\n
\n
\nThe centre is , so the y-coordinate is .
Q9·Challenging
A circle has centre and passes through the point . What is the radius of the circle?
Explanation
Apply the distance formula from to :\n
Q10·Challenging
A circle has centre and passes through . The equation of the circle is . What is ?
Explanation
Substitute into the equation:\n
\nSo and the equation is .
Q11·Challenging
The circle intersects the x-axis at two points. What is the larger x-intercept?
Explanation
Set :\n
\n
\n
\n
\n
\nThe larger x-intercept is .
Q12·Challenging
A circle has centre and passes through the origin. Which equation represents this circle?
Explanation
The radius is the distance from to :\n
\nThe equation is , which simplifies to .