Equations of Circles

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Equations of Circles

A circle with centre $(a,b)$ and radius $r$ has equation $(x-a)^2+(y-b)^2=r^2$ . Completing squares converts general quadratic coordinate equations into centre-radius form.

Worked reasoning

Compare with $(x-a)^2+(y-b)^2=r^2$ .
$r^2=25$ , so the positive radius is $r=5$ .

Key idea

Circle equations encode a fixed squared distance from a centre.

For $(x-3)^2+(y+2)^2=25$ , what is the radius?

Which statement best captures the central mathematical idea in Equations of Circles?

Circle equations encode a fixed squared distance from a centre.

In $(x-a)^2+(y-b)^2=r^2$ , the centre is $(-a,-b)$ .

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Equations of Circles, which move is most reliable?

Complete the square separately in $x$ and $y$ and compare with centre-radius form.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Equations of Circles?

In $(x-a)^2+(y-b)^2=r^2$ , the centre is $(-a,-b)$ .

Circle equations encode a fixed squared distance from a centre.

Complete the square separately in $x$ and $y$ and compare with centre-radius form.

Substitute a known point and verify its squared distance equals $r^2$ .

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