1. Equations of Circles

2. Centre-radius form of a circle
Center is at (h, k)
r is the radius of the circle
(h,k) r

3. General Form of a Circle
Expand the brackets and make the RHS equal to 0.
Every term is on the left side, equal to 0.

4. EX 1 Write an equation of a circle with center (3, -2) and a radius of 4.k r

5. EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10.k

6. EX 3 Write an equation of a circle with center (2, -9) and a radius of .k r

7. Opposite signs! ( , ) 6 -3 Take the square root! Radius 5
EX 4 Find the coordinates of the center and the measure of the radius.
Opposite signs!
( , ) 6 -3
Take the square root!
Radius 5

8. 5. Find the center, radius, & equation of the circle.
The center is
The radius is
The equation is
(0, 0) 12
x2 + y2 = 144

9. 6. Find the center, radius, & equation of the circle.
(1, -3)
The center is
The radius is
The equation is 7
(x – 1)2 + (y + 3)2 = 49

10. 7. Graph the circle, identify the center & radius.
(x – 3)2 + (y – 2)2 = 9
Center (3, 2)
Radius of 3

11. Converting from general form to centre-radius form
Move the x terms together and the y terms together.
Move C to the other side.
Complete the square (as needed) for x.
Complete the square(as needed) for y.
Factor the left & simplify the right.

12. 8. Write the centre-radius equation of the circle
8. Write the centre-radius equation of the circle. State the center & radius.
Center: (4, 0) radius: 3

13. Using TI Nspire CAS
In the calculator screen, select Menu, Algebra, Complete the square,
Type completeSquare(x2+y2-8x=7,x,y)
The calculator will return (x-4)2+y2=23
Equate the above expression to 0 and move 9 to the right-hand side.
Refer to Ti Nspire CAS handout how to sketch circles on your calculator.

17. 9. Write the centre-radius equation of the circle
9. Write the centre-radius equation of the circle. State the center & radius.
Center: (-2, 3) radius: 4

18. 10. Write the centre-radius equation of the circle
10. Write the centre-radius equation of the circle. State the center & radius.

19. 11. Write the general form of the equation of the circle given in centre-radius form.

20. Semicircles
Given the equation of a circle, we make y the subject.
Example: Write down the equation of the upper and lower semicircle, given
Therefore the equation of the upper semicircle is
And the equation of the lower semicircle is

22. Shading regions
Draw a circle and test with a point whether it is inside or outside.
In general is the region outside the circle without the boundary so a dotted line.
And is the region inside the circle, including the boundary; thus a continuous line.

23. Homework
Exercise 4D Q1 d, e Q2 a, d Q3 c, f Q4 b, d by hand; Q4 a, e on the calculator Q5 b Q6 b, d Q7a
Exercise 4E
Q 9
Q 14 a