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Introduction
A circle is the set of all points in a plane situated at a fixed distance from a fixed point in the plane. The fixed point in the process is called the centre of the circle. Distance from the centre of the circle to any point on the circle is called the circle’s radius. All equations related to circles represent the points located at the circumference or the circle’s boundary.
Cartesian equations in a circle can be algebraically determined by quickly drawing a circle using its centre and radius on the cartesian plane. There are different forms of circles like general, standard, parametric, and polar form.
General equation
For a circle, general equation is represented as x^2 + y^2 + 2ax + 2by + c = 0
Where,
x = point at the boundary
y = point at the boundary
a = constant
b = constant
c = constant
Centre of this circle = (-a, -b)
Radius of the circle = √(a^2 + b^2 – c)
Standard equation
The standard equation of a circle is represented as:
(x – a)^2 + (y – b)^2 = r^ 2
Where,
r = radius of the circle
(a,b) = centre of the circle
This equation is also termed as circle-radius form.
(x, y) are the point at the boundary or circumference of the circle.
Parametric equation
The general equation is
x^2 + y^2 + 2ax + 2by + c = 0
θ = angle between line joining boundary to the centre of the circle
Boundary = (x, y)
Centre of the circle = (a, b)
If the equation is x^2 + y^2 = r^2, then the parametric equation is given by:
x = r cosθ, y = r sinθ
If the standard equation was (x – a)^2 + (y – b)^2 = r^2,
The parametric equation is given by:
x – a = r cosθ, y – b = r sinθ
or x = a + r cosθ, y = b + r sinθ
Circle with centre (0,0) and radius r
By distance formula, standard equation is (x – a)^2 + (y – b)^2 = r^2
Here, a and b are the coordinates of the centre of the circle, hence both are zero.
(x – 0)^2 + (y – 0)^2 = r^2
x^2 + y^2 = r^2
Equation of circle with centre on X-axis
Consider (x,y) as arbitrary points on the circle’s circumference.
The circle’s centre is on the x-axis (-a,0) with radius r.
The equation is
x^2 + y^2 + 2ax + c = 0
Equation of a circle with centre on Y-axis
(x,y) are arbitrary points on the circumference of the circle. The circle’s centre is on the y-axis (0, -b) with radius r.
- The general equation is:
x^2 + y^2 + 2by + c = 0
2. The circle passing through the origin ,The equation becomes
x^2 + y^2 + 2ax + 2by = 0 and c=0
3. The equation for a circle passing through the origin that cuts the x-axis at (a,0) and the y-axis at (0,b) is:
x^2 + y^2 – ax -by = 0
4. The circle touches the Y-axis at origin and is centred at X-axis.
The equation is x^2 + y^2 – 2ax = 0
( 0,0) and (2a, 0) are the endpoints of the diameter of the circle.
5. The circle touching X-Axis at origin and centred at Y-Axis
The equation is x^2 + y – 2by = 0
(0,0) and (0, 2b) are the endpoints of the diameter.
Cartesian equation of a circle example
Example : Find the centre and the radius of the circle x^2 + y^2 + 8x + 10y – 8 = 0
Solution :
The given equation is: (x^2 + 8x) + (y^2 + 10y) = 8
On completing the squares within parenthesis,
(x^2 + 8x + 16) + (y^2 + 10y + 25) = 8 + 16 + 25
(x + 4)^2 + (y + 5)^2 = 49
(x – (-4))^2 + (y – (-5))^2 = 72
On comparing it with the general equation,
The centre of the given circle is (-4, -5), and the radius is 7
Answer: Centre = (-4, -5), and radius = 7
Conclusion
The circle radius is the distance from the circle’s centre to the boundary, so its values always remain positive. There are many different cartesian equations of a circle depending upon the circle’s coordinates. It helps to find the centre and radius of the circle in the cartesian plane. One form of equation can be turned into another form with the help of the formulae using the values given in the questions.
