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Before we discuss on the equation of circle, this is necessary to lean about Circle first. Circle is a set of all points that are equidistant from a fixed point within a plane. The fixed point is named as the centre of the circle. The distance between centre to a point on the circumference is named as the radius of circle. The general form of a circle when it is centred at the origin, is given as –

STANDARD FORM OF CIRCLE EQUATION

$\LARGE (x – a)^{2} + (y – b)^{2} = r^{2}$

GENERAL FORM OF CIRCLE EQUATION

$\LARGE x^{2} + y^{2} + Ax + By + C = 0$

Where,
a, b is the center,
r is the radius

In geometry, there are a plenty of facts associated with circles and their relationships with straight lines, angles, or polygons etc. Also, every circle has a plenty of properties that need to satisfy to make further calculations.

But, how the things will change if the circle is not centred at the origin. In this case, you need to make small adjustments to the formula. Don’t sweat, just minus the distance from x and y values and re-write the equation as given below –

With little algebra, you again can convert the formula into standard form by putting the values of h and k coordinates zero. This technique allows you to keep your circle again originate at the centre.

Instead of using the Pythagorean theorem to solve the circle above, the best idea is to follow trigonometry techniques. In this way, you could describe the parametric equation of a circle as needed. The parametric equation is useful for computer algorithms to draw circles or ellipses. A depth understanding of the topic and related concepts will surely help you to apply them in real-life situations.

Solved Examples

Question 1: If the center point and radius of a circle is given as (4, 5) and 7 respectively. Represent this as a circle equation ?

Solution:

Given parameters are
Center (a, b) = (4, 5); radius r = 7
The standard form of circle equation is,
(x-a)2 + (y-b)2 = r2
So, (x-4)2 + (y-5)2 = 72
So, (x-4)2 + (y-5)2 = 49