Loading....
Coupon Accepted Successfully!

 

Definition

A circle is the locus of a point which moves in a plane such that its distance from a fixed point is always a constant. The fixed point is called the center and the constant distance is called the radius of the circle.

Circle with center (h, k) and radius r

The equation of circle is (xh)2 + (yk)2 = r2. In particular, if the center is at the origin, the equation of circle is x2 + y2 = r2.

General equation of a circle

The general equation of circle is x2 + y2 + 2gx + 2fy + c = 0, where g, f, and c are constants. Therefore, coordinates of the center are (–g, –f) and radius = 68745.png.

 

Notes:
  • A general equation of second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 in xyrepresent a circle if
    • coefficient of x2 = coefficient of y2, i.e., a =b
    • coefficient of xy is zero, i.e. h = 0
  • Concentric circle: Two circles having the same center C (hk) but different radii r1and r2 respectively are called concentric circles.
  • Equation of circle passing through three given points: The general equation of circle
     
    x2 + y2 + 2gx + 2fy + c = 0 ...(1)
     
    contains three independent constants gfand c.
     
    If three points (x1y1), (x2y2), (x3y3) lie on the circle (1), their coordinates must satisfy its equation. Hence solving equations
     
    69731.png = 0 ...(2)
     
    69738.png = 0 ...(3)
     
    69627.png = 0 ...(4)
     
    we can find the values of gf, and c.
  • Cyclic quadrilateral: If all the four vertices of a quadrilateral lie on a circle, then the quadrilateral is called a cyclic quadrilateral. The four vertices are said to be concyclic.
  • Equation of circle having center (αβ) and touching the x-axis is (x – α)2 + (y – β)2β2.
  • Equation of circle having center (αβ) and touching the y-axis is (x – α)2 + (y – β)2α2.
  • Equation of circle having radius and touching both axes is (x – α)2 + (y – α)2α2.

Equation of circle on a given diameter

Let A(x1, y1) and B(x2, y2) be the end points of a diameter then (xx1) (xx2) + (yy1) (yy2) = 0 which is equation for circle in diametric form.
 
Equation of circle passing through two points A and B having least radius is the circle having AB as diameter.

Parametric form of circle

The parametric coordinates of any point on the circle (xh)2 + (yk)2 = r2 are given by (h + r cos θ, k + r sin θ), where θ is parameter (0 ≤ θ < 2π) then x = h + r cos θ and y = k + r sin θ.
 
In particular, coordinates of any point on the circle x2 + y2 = r2 are (r cos θ, r sin θ) (0 ≤ θ < 2π).




Test Your Skills Now!
Take a Quiz now
Reviewer Name