How is a circle different from a line?
Equation of a circle is more involved, and more complex
than a straight line.
A stone is dropped into a pond, creating
a circular ripple. The radius of the ripple
increases by 4 cm/s. Determine an
equation that models the circular ripple,
10s after the stone is dropped.
40cm
Moving at 4 cm/s, 10 s after the ripple is created, it has
moved 40cm.
r = 40cm
x2 + y2 = r2 - Equation of a circle
x2 + y2 = 402
x2 + y2 = 1600
So, the equation of the circular ripple is x2 + y2 = 1600.
A circle is defined by the equation x2 + y2 = 25. Sketch a
graph of this circle.
To find the x-intercepts, let y = 0:
x2 + y2 = 25
x2 + (0)2 = 25
x2 = 25
x = ±5
The x-intercepts are located at (5, 0) and (-5, 0).
To find the y-intercepts, let x = 0:
x2 + y2 = 25
(0)2 + y2 = 25
y2 = 25
y = ±5
The y-intercepts are located at (0, 5) and (0, -5).
The x-intercepts
are located at (5,
0) and (-5, 0).
The y-intercepts
are located at (0,
5) and (0, -5).
A circle has its centre at (0, 0) and passes through point (8, -6).
a) Determine the equation of the circle.
b) Determine the other endpoint of the diameter through (8,-6).
a) x2 + y2 = r2
(8)2 + (-6)2 = r2
64 + 36 = r2
r2 = 100
Therefore, the equation of the circle is x2 + y2 = 100.
b) We can figure out the other endpoint through reasoning. If
there is one point on the circle at (8, -6), and the circle is
symmetric, there must be another point at the other end of the
diameter that is (-8, 6), because it’s radius is also 10.
• Every point on the circumference of a circle is the same
distance from the centre of the circle
• Once you know one point on a circle with centre (0, 0),
you can determine other points on the circle using
symmetry
• If (x, y) is on a circle with centre (0, 0), then so are (-x,
y), (-x, -y), and (x, -y).