Problems from Equation of the circumference II: general equation

Considering the circumference $$x^2+y^2+2x-4y+4=0$$, find its radius and its center.

See development and solution

Development:

In this case we have $$A=-2$$, $$B=4$$ and $$C=-4$$. Therefore, the center will be $$$\Big(\dfrac{-A}{2},\dfrac{-B}{2}\Big)=\Big(\dfrac{2}{2},\dfrac{-4}{2}\Big)=(1,-2)$$$ and the radius will be $$$r=\sqrt{\Big(\dfrac{A}{2}\Big)^2+\Big(\dfrac{B}{2}\Big)^2-C}=\sqrt{\Big(\dfrac{-2}{2}\Big)^2+\Big(\dfrac{4}{2}\Big)^2+4}=$$$ $$$=\sqrt{1+4+4}=\sqrt{9}=3$$$

Solution:

Center $$(1,-2)$$ and radius $$3$$.

Hide solution and development

Find the equation of the circumference with center $$(3, 4)$$ and radius $$2$$.

See development and solution

Development:

$$$(x-3)^2+(y-4)^2=2^2 \Rightarrow x^2+y^2-6x-8y+9+16-4=0 \Rightarrow $$$ $$$\Rightarrow x^2+y^2-6x-8y+21=0$$$

Solution:

$$(x-3)^2+(y-4)^2=4$$ or also $$x^2+y^2-6x-8y+21=0$$

Hide solution and development
View theory