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Equation of a circle

Equation of a circle

The equation of a circle, with centre $C(A,B)$ and radius $r$ is $$(x-a)^2 + (y-b)^2 = r^2$$

We can quickly derive this formula by considering the digram.Deriving the equation of a circle

We take an arbitrary point on the perimeter of the circle and consider the right angled triangle shown. The length of the horizontal edge of the triangle is $(x-a)$ the length of the vertical edge is $(y-b)$. Applying Pythagoras's theorem yields $(x-a)^2 + (y-b)^2 = r^2$

Equation of a circle 1

Determine the equation of a circle with centre $(7,3)$ and radius $5$
solution - press button to display
Recall that the equation of a circle is $(x-a)^2 + (y-b)^2 = r^2$. Substitution of the values from the question gives $$ (x-7)^2 + (y-3)^2 = 5^2 $$

Equation of a circle 2

Determine the centre and radius of the circle with equation $$ x^2 + 8x + y^2 - 4y = 16 $$
solution - press button to display
To proceed complete the square on both the $x$-terms and the $y$-terms. $$ x^2 + 8x + y^2 - 4y = 16 \Leftrightarrow (x+4)^2 - 16 + (y-2)^2 - 4 = 16 $$ rearranging this into the standard form yields $$ (x+4)^2 + (y-2)^2 = 36 $$ The centre is therefore $(-4,2)$ and the radius $6$.