How do you graph 4x^2+4y^2+24x+32y+75=0?

Oct 21, 2016

Draw a circle with center at $\left(- 3 , - 4\right)$ and a radius of $2 \frac{1}{2}$ units.

Explanation:

$4 {x}^{2} + 4 {y}^{2} + 24 x + 32 y + 75 = 0$

$\rightarrow 4 \left({x}^{2} + 6 x \textcolor{w h i t e}{\text{XXX"))+4(y^2+8ycolor(white)("XXX}}\right) = - 75$

$\rightarrow 4 \left({x}^{2} + 6 x + {3}^{2}\right) + 4 \left({y}^{2} + 8 y + {4}^{2}\right) = - 75 + 4 \cdot \left({3}^{2}\right) + 4 \cdot \left({4}^{2}\right)$

$\rightarrow 4 {\left(x + 3\right)}^{2} + 4 {\left(y + 4\right)}^{2} = 25$

$\rightarrow {\left(x + 3\right)}^{2} + {\left(y + 4\right)}^{2} = \frac{25}{4} = {\left(\frac{5}{2}\right)}^{2}$

which is the standard form for a circle with center $\left(- 3 , - 4\right)$ and radius $\frac{5}{2} = 2 \frac{1}{2}$

graph{4x^2+4y^2+24x+32y+75=0 [-7.61, 4.88, -6.81, -0.566]}