# How do you write the equation of a circle centered at (-5,2) and contains the points (7,3)?

Jan 8, 2017

${\left(x + 5\right)}^{2} + {\left(y - 2\right)}^{2} = 145$

#### Explanation:

The formula of a circle centred at the origin is ${x}^{2} + {y}^{2} = {r}^{2}$ because you are using the principle of Pythagoras on a triangle.

When the centre of the circle is not at the origin you mathematically move it back to centre.

So we have

${\left(x + 5\right)}^{2} + {\left(y - 2\right)}^{2} = {r}^{2}$

Now we need to find the length $r$ and this is achieved by calculating distance from centre to the given point using the well known
${a}^{2} + {b}^{2} = {c}^{2}$ for a triangle

$r = \sqrt{{\left(- 5 - 7\right)}^{2} + {\left(2 - 3\right)}^{2}} = \sqrt{{12}^{2} + {1}^{2}} \implies {r}^{2} = 145$

So we now have:

${\left(x + 5\right)}^{2} + {\left(y - 2\right)}^{2} = 145$
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$\textcolor{b l u e}{\text{Comment:}}$

x_("centre")+5 = -5+5=0 larr" moved to origin"
y_("centre")-2 = +2-2=0 larr" moved to origin"