Answer to Question #156829 in Analytic Geometry for Hillary

Question

Find the radius and centre of the circle:
(a) x^2 + y^2 - 2x + 4y + 1 = 0

Answer 1

Given Equation :

$x^2+y^2-2x+4y+1=0$

which is the form of general equation of circle :

$x^2+y^2+2gx+2fy+c=0$

with center at $(-g,-f)$ and radius being $\ r=\sqrt{g^2+f^2-c}$

In given equation,

$g=\frac{-2}{2}=-1\implies -g=1$

And $f=\frac{4}{2}=2\implies -f=-2$

And $c=1$

So, center of circle is $(-g,-f)\equiv(1,-2)$

And radius of circle will be $\ r=\sqrt{g^2+f^2-c}=\sqrt{(-1)^2+2^2-1}=\sqrt{4}=2$ units.