Circle - Practice Questions & MCQ

Circle

Definition

A circle is the locus of a moving point such that its distance from a fixed point is constant.

The fixed point is called the centre (O) of the circle and constant distance is called its radius (r)

Equation of circle

Centre-Radius Form

The equation of a circle with centre at C (h,k) and radius r is (x - h)² + (y - k)² = r²

Let P(x, y) be any point on the circle. Then, by definition, | CP | = r.

Using the distance formula, we have

If the centre of the circle is the origin or (0,0) then equation of the circle becomes

General Form:

The equation of a circle with centre at (h,k) and radius r is 

This is known as the general equation of the circle.

To get radius and centre if only the equation of the circle (ii) is given:

Compare eq (i) and eq (ii)

h = - g, k = - h  and c = h² + k² - r²

Coordinates of the centre  (-g,-f)

Radius =

Nature of the Circle

For the standard equation of a circle x²+y²+2gx+2fy+c=0 whose radius is given as 

Now the following cases arise

1. If g²+f²-c > 0, then the radius of the circle will be real. Hence, the circle is a real circle.

2. If g²+f²-c = 0,  then the radius of the circle will be real (=0). Hence, the circle is a Point circle because the radius is 0.

3. If g²+f²-c < 0, then the radius of the circle will be imaginary. Hence, the circle is an imaginary circle.