Different forms of the equation of circles

Different Forms of the Equation of a circle

(a) (x - α)² + (y - α)² = α² is a circle with centre (α, α) and radius = α touching both the axes.

(b) (x - α)² + (y - β)²  = β² touches the x – axis.

(c) (x - α)² + (y - β)² = α² touches the y - axis.

(d) x² + y² - αx - βy = 0 is with centre (α/2,β/2) and

radius = . It passes through (0, 0), (α, 0) and (0, β).

Equation of the tangent to the circle

x² + y² + 2gx + 2fy + c = 0 at the point (x₁, y₁) is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

and that of the normal is y – y₁ =

In particular to the circle x² + y² = r² at (x₁, y₁), the equation of tangent is

xx₁ + yy₁ = r₂ and the normal .

Normal to the circle always passes through the centre.