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CIRCLE

A circle is the locus of a point which moves in a plane so that it is always at a constant distance from a fixed point in that plane. The fixed point is called the centre and the constant distance is called the length of the radius of the circle.

EQUATIONS OF A CIRCLE

1. THE CENTRE-RADIUS FORM

To find the equation of a circle with centre at (h,k) and radius r

Every point on the circle should be at a distance r from (h,k). This implies that if (x,y) is any point on the circle then by the distance formula.

 

(x-h)2 + (y-k)2 = r2

This is the required equation.

2. THE GENERAL EQUATION OF A CIRCLE

 

x2+y2+2gx+2fy+c = 0

This equation is the general equation of the circle, and it represents a circle with centre (-g, -f) and radius (g²+f²-c)^1/2

Note:

• If g²+f²-c < 0 then the circle is imaginary.
• If g²+f²-c = 0 the circle reduces to a single point (-g, -f).
• When g²+f²-c > 0 do we get a real circle.

3. THE TWO POINT FORM

The equation of the circle with diameter as AB where A is the point (x₁,y₁) and B is (x₂,y₂) is of the form

 

(x-x1)(x-x2)+(y-y1)(y-y2) = 0

We know that the angle in a semicircle is a right angle(refer figure)

So if P is any point on the circle, then the angle subtended by AB at P should be 90^O.

Thus, the (slope of AP). (Slope of PB) = -1.

=> {(y-y₁)/ (x-x₁)} .{ (y-y₂)/(x-x₂)} = -1

=> (x-x₁)(x-x₂) + (y-y₁)(y-y₂) = 0

GEOMETRICAL CONDITIONS REQUIRED TO DETERMINE A CIRCLE

Equations of a circle, both the standard form and the general form, contain three independent arbitrary constants. The circle is completely determined if we know the values of these three constants. In  order to be able to do this we need three conditions or information about the circle. The most  common of the three conditions are

1. Three given points through which the circle passes.

2. Two given points on the circle and a line on which the circle lies.

3. The central line, point and radius etc..

if the three conditions as any one of the above are states then we can find the equation of the required circle by,

1. Let the required equation of the circle be  x²+y²+2gx+2fy+c = 0

2. Use the three given conditions to establish three equations in g, f and c.

3. Solve these three equations in algebra for g,f and c and obtain their values.

4. Substitute the values of g, f and c in the general equation of a circle (x²+y²+2gx+2fy+c = 0). This gives the required equation.