Intermediate 2nd year Maths 2b circles

Intermediate 2nd year

Maths 2b circles

Introduction :

 A circle is the set of points in a

Plane such that there are equidistant form a fixed point lying in the plane. 

      The fixed point is called center is denoted by " C ". And the distance form the center to a point on the circle is called radius is denoted by " r ".

Note :

1) the equation of the circle with center O

(0, 0) & radius " r " is x² + y² = r²

2) the equation of the circle with center (h, k) 

& radius " r " is given by (x-h)² + (y-k)² = r²

3) the general equation if the second degree 

ax² +2hxy + by² +2gx + 2fy + c = 0

Where the coefficients a, h, b, g, f and c are real numbers, represents a circle if and only if 

i) a = b ≠ 0  (x² h coefficient = y² coefficient) 

ii) h = 0 (xy coefficients = 0) 

iii) radius ≥ 0 (g² + f² - ac ≥ 0) 

4) the general equation of the circle is 

x² + y² + 2gx + 2fy + c = 0

5) the center of the circle x² + y² +2gx + 2fy + c = 0 is  C = (-g, -f) 

6) the radius if the circle x² + y² +2gx + 2fy + c = 0 is r = √(g² + f² - c) 

7) if g² + f² - c = 0 then x² + y² +2gx + 2fy + c = 0 represents a point circle. In this case center itself is the point cricle. 

8) the equation of the point circle having the center at the origin is x² + y² = 0.

9) the equation of a circle passes through (0, 0) will be in the form  x² + y² +2gx + 2fy = 0

10) the equation of a circle having the center on x-axis will be in the form of x² + y² +2gx + c = 0

11) the equation of a circle having the center on y-axis will be in the form of x² + y² +2fy + c = 0

Concentric circles :

1) two or more circlee are said to be concentric, if there center are same and radius are diffferent. 

2) the equation if circle concentric with the circle x² + y² +2gx + 2fy + c = 0 will be in the form of x² + y² +2gx + 2fy + c' = 0

Where C' is the constant. 

3) if the radius of the circle if one then it is called unit circle. 

4) if g² - c > 0 then the intercept made on the x-axis by the circle x² + y² +2gx + 2fy + c = 0 is 2√(g²-c).

5) if f² - c > 0 then the intercept made on y-axis by the circle x² + y² +2gx + 2fy + c = 0 is 2√(f² - c). 

6) if g² - c = 0 

The x-axis touches the circle in two Co-incident  points. 

7) if f² - c = 0 

The y-axis touches the circle in two Co-incident  points 

8) if g² - c > 0 then the circle doesn't meet the x-axis. 

9) if f² - c > 0 then the circle doesn't meet the y-axis. 

Secant chrod :

If A & B are two distant points on a circle then 

1) then   through A & B is calle a secant as shown fig i) below

2) the segment  , the join A & B is called a chord. And the length of the chord is AB as shown in above fig ii) 

Note :

1) The equation of the circle whose diameter extremeties are (x₁, y₁) & (x₂, y₂) is

(x-x₁) (x-x₂)+ (y-y₁) (y-y₂) = 0

The parametric equation of a circle with center (h,k)  and radius r > 0 is 

x = h+rcosθ, y = k+rsinθ

x = -g + rcosθ, y = -f + rsinθ

Where  0<θ<2π