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IntroductionOne 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 CircleThe 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 problemType 1: Find the equation of a circle given its centre and its radius. Example 1Find 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 2Find 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: Trick: If (0,0) is on a circle then c = 0 automatically. Example 3Find 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:
Example 4Find the equations of the circle of centre on the X-axis, radius 2 and (0,1) on the circle.Solution Example 5Find 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.
Example 6Find 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 7Find 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 |