About Equation of a circle formula
The standard equation of a circle is given by: (x-h)2 + (y-k)2 = r2
Where (h,k) is the circle's centre coordinates and r is the radius. For more Maths formulas click on the main page.
What is the Equation of a Circle?
A circle is a closed curve that is drawn from a set point called the centre and has the same distance between all of its points. The equation for a circle with a radius of r and a centre of (h, k) is: (x-h)2 + (y-k)2 = r2, This is the standard form of the equation.
The circle's equation, with the origin at the centre
x2+y2= a2, Where “a” is the radius of the circle.
Let P(x, y) be any point on the circle and C(h, k) be the circle's centre.
As a result, a circle's radius is CP.
Using the formula for distance, (x-h)2 + (y-k)2 = CP2
Let’s suppose the radius be ‘a’.
Thus, the Equation of a circle with centre (h, k) and radius ‘a’ is,
(x-h)2 + (y-k)2 = a2, This is known as the conventional form for a circle equation.
Circle Equation in General Form
Any sort of circle has the following generic equation: x2 + y2 + 2gx + 2fy + c = 0, for all values of g, f and c.
By adding g2 + f2 on both sides of equation gives,
x2 + 2gx + g2+ y2 + 2fy + f2= g2 + f2 − c ………………(1)
Since, (x+g)2 = x2+ 2gx + g2 and (y+f)2 =y2 + 2fy + f2 substituting the values in equation (1), we have
(x+g)2+ (y+f)2 = g2 + f2−c …………….(2)
On comparing (2) with (x−h)2 + (y−k)2 = a2, where (h, k) is the centre and ‘a’ is the radius of the circle.
h=−g, k=−f, a2 = g2+ f2−c
Therefore, x2 + y2 + 2gx + 2fy + c = 0, represents the circle with centre (−g,−f) and radius equal to a2 = g2+ f2−c.
If g2 + f2 > c, then radius of circle is real.
If g2 + f2 = c, The circle's radius is then 0, indicating that it is a point that coincides with the centre. A point circle is one such sort of circle.
g2 + f2 less than c, The circle's radius then becomes imaginary. It is thus a circle with a real centre and an imagined radius.
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