Circle, Definition, Equation of circle whose center and radius is known, General equation of a circle, Equation of circle passing through three given points, Equation of circle whose diameters is line joining two points (x1, y1) & (x2,y2), Tangent and Normal to a given circle at given point.

1. Dr. Ramkrishna Singh Solanki
Asst. Professor: Mathematics and Statistics
College of Agriculture Balaghat
1
Unit: Circle

2. Definition
•A circle is defined as the locus of a point which moves is a plane such that its
distance from a fixed point in that plane is always constant.
•The collection of all the points in a plane, which are at a fixed distance from a fixed
point in the plane, is called a circle.
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3. Basics

4. Day to day life example

5. Topic: Equation of a circle whose center and radius are known

6. Topic: Equation of a circle whose center and radius are known
Problem 1: Find out the equation of a circle when center and radius are known.
Problem 2: Find out the center and radius when equation of a circle is known.

7. (h, k)
Y
X
r
Center

8. Q.
…(2)

9. Topic: General equation of a circle

10. → x2 + y2 +2gx +2fy + c = 0 (1)
→ x2 + y2 +2gx +2fy + c + g2 + f 2 – c = g2 + f 2 – c (2)
→ (x2 +2gx + g2) + (y2 +2fy + f 2) = g2 + f 2 – c (3)
→(x + g)2 + (y + f)2 = g2 + f2 – c (4)
→{x – (- g)}2 + {y – (- f)}2 = (5)
(x - h)2 + (y - k)2 = r 2 (compare with standard equation of circle)
we get
 2
2
2
c
f
g 

Topic: General equation of a circle
The general form of the equation of a circle is
x2 + y2 + 2gx + 2fy + c = 0
Centre = (-g, -f) and radius = , where g, f and c are constants.
(x2 +2gx + g2) = (x + g)2
(a2 +2ab + b2) = (a + b)2
c
f
g 2
2


c
f
g
redius
r
f
g
center
k
h 






 2
2
)
,
(
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,
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11. 




12. Given equation
Compare with general equation
We get g
We get f
We get c

14. Topic: Equation of circle whose diameters is line joining two points (x1, y1) & (x2,y2)
The equation of the circle drawn on the straight line joining two given points (x1, y1) and (x2,
y2) as diameter is
(x – x1)(x – x2) + (y – y1)(y – y2) = 0
Q. Find the equation of the circle, the coordinates of the end points of whose
diameter are (-1, 2) and (4, -3).
Sol. Here, x1 = -1, y1 = 2 and x2 = 4, y2 = -3, so the equation of the required circle is
→ (x – x1)(x – x2) + (y – y1)(y – y2) = 0
→ (x +1)(x – 4) + (y – 2)(y + 3) = 0
→ x2 - 4x +x -4 + y2 + 3y – 2y – 6 = 0
→ x2 + y2 - 3x + y – 10 = 0

15. How to solve three linear equations with three unknowns?
(1)
(2)
(3)
(4)
(5) (6)
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( )
(7)
(8)
(4)
(5)

16. Topic: Equation of a circle passing through three given points
25
8
17
(x, y)=
(x, y)=
(x, y)=
General equation of a circle

17. Topic: Tangent and Normal to a given circle at given point
स्पर्श-रेखा
प्रसामान्य
स्पर्शज्या

18. 1. Angle of bisectors between two lines
2. Condition of tangency of a line y = mx + c to the given circle x2 + y2 = a2