Circle

A circle is a closed curve such that every point on the curve is at a constant distance from a fixed point. Circle may also be defined as a 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

1. Centre at the 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
Squaring both sides we have,
OP²= r²
or, (x - 0)² + (y - 0)² = r²
or, x² + y² = r²
This relation is true for any point P (x , y) on the circle. So, it is the equation of the circle.

2. Centre at any point (Central form)
Let Q (h, k) be the centre and r be the radius of the circle.
Let P (x, y) be any point on the circle.
Then, QP = r
Squaring both sides we have,
QP² = r²
or, (x - h)² + (y k)² = r²
This is the equation of the circle.

3. 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, then \(\angle\)APB is a right angle.
Now,
Slope of AP = \(\frac {y - y_1}{x - x_1}\)
Slope of BP = \(\frac {y - y_2}{x - x_2}\)
Since,
AP is perpendicular to BP, the product of their slopes must be -1.
Hence,
\(\frac {y - y_1}{x - x_1}\) . \(\frac {y - y_2}{x - x_2}\) = -1
or, (y - y₁) . (y - y₂) = - (x - x₁) . (x - x₂)
or, (x - x₁) . (x - x₂) +(y - y₁) . (y - y₂) = 0
This relation is satisfied by any point on the circle. So, it is the equation of the circle.

4. General equation of the circle
Let Q (h, k) be the centre and r be the radius of the circle.
Let P (x, y) be any point on the circle.
Then,
QP = r
Squaring on both sides,
QP² = r²
or, (x - h)² + (y - k)² = r²
or, x² - 2hx + h² + y² - 2ky + k² = r²
or, x² + y² - 2hx - 2ky + h² + k² - r² = 0
This is the equation of a circle having centre at the point (h, k) and radius r.
Putting -2h = 2g, -2k = 2f and h² + k² - r² = c we have,
x² + y² + 2gx + 2fy + c = 0
This is general equation of the circle.

Note: The general equation of a circle is: x² + y² + 2gx + 2fy + c = 0.
Comparing this equation with the general equation of second degree
ax² + 2hxy + by² + 2gx + 2fy + c = 0 we have,
- coefficient of x² and y² are equal
- coefficient of xy is zero
Hence,
The general equation of second degreeax² + 2hxy + by² + 2gx + 2fy + c = 0 represents a circle if
- coefficients of x² and y² are equal i.e. a = b
- coefficient of xy is zero i.e. h = 0

Centre and radius of a circle

The general equation of a circle is:

x² + y² + 2gx + 2fy + c = 0

or, x² + 2gx + y²+ 2fy = -c

or, x² + 2gx + g² + y² + 2fy + f² = g² + f² - c

or, (x + g)² + (y + f)² = (\(\sqrt {g^2 + f^2 - c}\))²

Comparing this equation with (x - h)² + (y - k)² = r², we have

h = -g, k = -f and r = \(\sqrt {g^2 + f^2 - c}\)

Hence,

Centre of the circle (h, k) = (-g, -f)

Radius of the circle (r) = \(\sqrt {g^2 + f^2 - c}\)

Equation of a circle passing through three points

Let, A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) be three points of a circle.

Let, P (h, k) be the centre of the circle.

Then,

AP = BP = CP

or, AP² = BP² = CP²

By using distance formula,

We have,

AP²= (x₁ - h)² + (y₁ - k)²

BP²= (x₂ - h)² + (y₂ - k)²

CP²= (x₃ - h)² + (y₃ - k)²

Taking AP² = BP² we have,

(x₁ - h)² + (y₁- k)² = (x₂ - h)² + (y₂ - k)² .....................................................(i)

Taking BP² = CP² we have,

(x₂ - h)² + (y₂- k)² = (x₃ - h)² + (y₃ - k)² .....................................................(ii)

By solving (i) and (ii) we will get the values of h and k. Hence, we get centre P (h, k) of the circle. Then length of AP or BP or CP gives the radius r of the circle.

Now, Putting the values of h, k and r in (x - h)² + (y - k)² = r² we get the required equation of the circle.

Equation of a circle in particular cases

1. When a circle touches the X - axis

Let the centre of a circle be C(h, k) and radius r. If this circle touches X- axis and the circle is in the first or second quadrant, then r = k.
If the circle is in the third or fourth quadrant, then r = -k.
Now,
The equation of a circle touching the X- axis is
(x - h)² + (y - k)² = r²
or, (x - h)² + (y - k)²= k²

2. When a circle touches the Y- axis

Let C (h, k) and r be the centre and radius of a circle.
If this circle touches the Y- axis and it is in the first or fourth quadrant, then r = h.
If the circle is in the second or third quadrant, then r = -h.
Now,
The equation of a circle touching the Y- axis is
(x - h)² + (y - k)² = r²
or,(x - h)² + (y - k)² = h²

3. When a circle touches both the positive axes

Let C(h, k) and r be the centre and radius of a circle, if this circle touches both the positive axes, then h = k = r.
Now,
The equation of the circle is:
(x - h)² + (y - k)² = r²
or,(x - h)² + (y - k)² = h²
or,(x - k)² + (y - h)² = k²
or,(x - r)² + (y - k)² = k²