# Equation of family of circles

1. The equation of circle with centre A (h, k) and radius AP = r : (x - h)^{2} + (y - k)^{2} = r^{2}

2. The equation of circle with centre at origin 0(0, 0) and radius = r:

x^{2} + y^{2} = r^{2}

3. The equation of circle with centre (-g, -f) and radius = r :

x^{2} + y^{2} +2gx +2fy +c = 0, where c = g^{2} + f^{2} - r^{2}

This is also known as general form of equation of circle. For this circle:

(i) Centre is ( - g , -f)

(ii) Radius = r =

# Different types of Circles

**(i) Centered at X-axis:**

=> Y-coordinate of centre 0, i.e., f = 0

=> centre is (-g , 0)

=> x^{2} + y^{2} + 2gx + c = 0

**(ii) Centered at Y-axis:**

=> X-coordinate of centre 0, i.e., g = 0

=> centre is. (0,-f)

=> x^{2} + y^{2} + 2fy + c = 0

**(iii) Centered at origin:**

=> center is (-g, -f) (0, 0)

=> x^{2} + y^{2} + c = 0 is the equation of circle. (c = -r2)

**(iv) Circle passing through origin:**

i.e., (0, 0) must lie on the circle or (0, 0) satisfies the equation of circle.

=> c = 0

=> x^{2} + y^{2} + 2gx + 2fy = 0

**Note:**

A circle x^{2} + y^{2} - ax - by =0 passes through origin and cuts X-axis at (a, 0) and Y-axis at (0, b).

**(v) Circle centered at X-axis and touching Y-axis at origin:**

x^{2} + y^{2} + 2gx = 0 is the circle.

=> (0, 0) and (-2g, 0) are end points of diameter.

**(vi) Circle centered at Y-axis and touching X-axis at origin:**

x^{2} + y^{2} + 2fy = 0 is the circle.

=> (0, 0) and (0, -2f) are end points of diameter.

**(vii) Circle touching both the axes:**

The centre of circle of radius r touching both axes OX and OY is:

(Â± r, Â± r)

(x r)^{2} + (y r)^{2} = r^{2}

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**(viii) Equation of circle in diametric form:**

(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) = 0 is the equation of circle,

where (x_{1}, y_{1}) and (x_{2}, y_{2}) are the end points of any diameter.

**(ix) Parametric form of circle: **

Consider the circle (x - h)^{2} + (y - k)^{2} = r^{2} centered at A (h, k) and of radius r.

Let P (x, y) be any point on this circle. Say AP has a slope of tan (). Thus coordinates of P can be expressed as:

* *(h + projection of AP on x axis)

(k + projection of AP on Y axis)

These two equations represent the coordinates of any point on the circle in terms of the parameter Î¸.

Thus for a circle centered at origin, the parametric form of equations are: