The Circle

Equation of a Circle


Circle

Consider a fixed point. Then, a circle is a closed curve around the point such that every point on the curve is at a constant distance from that point. In terms of locus, a circle may be defined as the locus of a point which moves so that its distance from a fixed point is constant.

The fixed point is called the centre and the constant distance is called the radius of the circle.

Equation of a Circle

Centre at the origin (Standard form)

Equation of a circle with centre at origin (Standard Form)

Let O(0,0) be the centre and r be the radius of the circle. Let P(x,y) be any point on the circle. Then, OP=r or, OP2=r2 x2+y2=r2 This relation is true for any point P(x,y) on the circle. Hence it is the equation of the circle.

Centre at any point (Central form)

Equation of a circle with centre at any point (Central Form)

Let C(h,k) be the centre and r be the radius of the circle. Let P(x,y) be any point on the circle. Then, CP=r or, CP2=r2 (xh)2+(yk)2=r2 This is the equation of the circle. This relation can be used to find the equation of the circle for the following particular cases.

Equation of the circle touching the x-axis

Equation of a circle touching the x-axis

Let (h,k) be the centre of the circle. If the circle touches the x-axis, then the radius of the circle (r)=k. Hence, the equation of the circle touching the x-axis is (xh)2+(yk)2=k2

Equation of the circle touching the y-axis

Equation of a circle touching the y-axis

Let (h,k) be the centre of the circle. If the circle touches the y-axis, then the radius of the circle (r)=h. Hence, the equation of the circle touching the y-axis is (xh)2+(yk)2=h2

Equation of the circle touching both axes

Equation of a circle touching both axes

Let (h,k) be the centre of the circle. If the circle touches both axes, then the radius of the circle (r)=h=k. Hence, the equation of the circle touching both axes is (xh)2+(yh)2=h2

General equation of the circle

Consider an equation x2+y2+2gx+2fy+c=0 in which the coefficients of x2 and y2 are equal, each being unity and there is no term containing xy. The given equation may be written as x2+y2+2gx+2fy=c x2+2gx+g2+y2+2fy+f2=g2+f2c (x+g)2+(y+f)2=(g2+f2c)2 __(1) which is in the form of (xh)2+(yk)2=r2, hence these equation represents a circle. The equation of a circle in this form is called the general equation of the circle.

Comparing equation (1) with the equation of a circle (xh)2+(yk)2=r2 we have h=g, k=f and r=g2+f2c. Hence, the equation x2+y2+2gx+2fy+c=0 represents a circle whose centre is at (g,f) and radius equal to g2+f2c.

  • If g2+f2c>0, the radius is real, hence the equation gives a real geometric locus.
  • If g2+f2c=0, the radius is zero. In this case, the circle is called a point circle.
  • If g2+f2c<0, the radius is imaginary. In this case, we say that the equation represents a circle with a real centre and an imaginary radius.

The general equation of second degree ax2+2hxy+by2+2gx+2fy+c=0 represents a circle if a=b (coefficients of x2 and y2 are equal) and h=0 (coefficient of xy is zero).

Circle with a given diameter (Diameter form)

Let A(x1,y1) and B(x2,y2) be the ends of a diameter of a circle. Let P(x,y) be any point on the circle. Join AP, BP and AB. Since AB is a diameter of the circle, APB is a right angle.

Equation of a circle with a given diameter (or end points) (Diameter form)

Slope of AP=yy1xx1 Slope of BP=yy2xx2 Since APBP, the product of their slopes is 1. yy1xx1.yy2xx2=1 or, (xx1)(xx2)+(yy1)(yy2)=0 This equation is satisfied by any points of the diameter of the circle. Hence, it is the equation of the circle.


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