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The Circle

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Introduction

One of the most difficult types of problem on the circle is to find its equation given certain information. The information can often be hard to understand and tricky to handle.

The Equation of a Circle

The equation of a circle centre (-g, -f) and radius r is:
where

It is important to note:
1. You must remember this formula.
2. You must not confuse (x,y), a point on the circle, with (-g ,-f), the centre of the circle.
3. What distinguishes one circle from another are the values of g, f, c. So to determine the equation of a circle you need to find g, f, c. There may, however be cases where the equation can be found without finding g, f, c.

Types of problem


Type 1: Find the equation of a circle given its centre and its radius.

Example 1

Find the equation of the circle centre (-2,3) and radius 4.
Solution

Type 2: Find the equation of a circle with 2 points on the ends of a diameter. There are 2 methods.

Example 2

Find the equation of the circle with a(-1,2) and b(3,-4) as points on the ends of a diameter.
Solution

Type 3: Find the equation of a circle given 3 points on it. This is one of the most tedious types of circle problem.
Steps:
  • Plonk in each point into the equation in turn, tidy up and label.
  • Solve them simultaneously by eliminating c from two pairs of equations.
  • If you spot any shortcuts, take them.
    Trick: If (0,0) is on a circle then c = 0 automatically.

    Example 3

    Find the equation of the circle with a (6,2), b (5,3) and d (3,-1) on it.
    Solution

    Type 4: Find the equation of a circle if you are told that the centre is on a line. You need 4 tricks:

    Trick 1
    The centre is on the X-axis
    (f = 0)

    Trick 2
    The centre is on the Y-axis
    (g = 0)

    Trick 3
    The centre is on a line ax + by + k = 0. Plonk the centre (-g, -f) into this equation. So if you are told that the centre of a circle is on a straight line 2x - 3y - 7 = 0 just plonk (-g, -f) into this equation to get -2g + 3f - 7 = 0.

    Trick 4
    Be careful. If you are told that a point is on a circle then plonk the point into the equation of the circle:

    Example 4

    Find the equations of the circle of centre on the X-axis, radius 2 and (0,1) on the circle.
    Solution

    Example 5

    Find the equation of the circle with centre on the line x - y = 0, radius 5 and (2,1) on the circle.
    Solution

    Type 5: Find the equation of a circle touching one or both axes. We look at 3 tricks:
    When you are told that a circle touches a line L:
    1. Draw the circle and mark its centre
    2. Draw the line L and mark the point of contact, a.
    3. Draw the line K (The Intelligent Line) joining the centre to the point of contact. It is always perpendicular to L.
    Trick 1: If a circle touches the X-axis
    Trick 2:
    If a circle touches the Y-axis

    Trick 3:
    If a circle touches both axes
    + if the circle is in the first or third quadrants.
    - if the circle is in the second or fourth quadrants.

    Example 6

    Find the equation touching both axes and passing through the point (2,4).
    Solution

    Type 6: Find the equation of the circle which intercepts the axes.
    Trick: When you are told that a line L intercepts (cuts) a circle always
    (i) Draw the circle and mark the centre o.
    (ii) Draw the line L and mark the two points of contact a and b.
    (iii) Draw the line K (The Intelligent Line) from centre o perpendicular to chord ab. This line always bisects ab.
    (iv) Join o to a and b to make two right-angled triangles. Pythagorus holds for each of these triangles.

    Example 7

    Find the equation of the circle passing through the origin and making an intercept of 6 units on the +X-axis and an intercept of 8 units on the +Y-axis.
    Solution



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