Question

what is the value of a for the following circle in general form?
a = _________
$x^{2}+y^{2}+ax + by + c = 0$

Explanation:

Step1: Find the center and radius of the circle

From the graph, the center of the circle is \((4,0)\) and the radius \(r = 2\).

Step2: Write the standard form of the circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius.
Substituting \(h = 4\), \(k = 0\), and \(r = 2\) into the formula: \((x - 4)^2+(y - 0)^2=2^2\).
Expanding \((x - 4)^2+y^2=4\): \(x^{2}-8x + 16+y^{2}=4\).

Step3: Convert to general form

Rearrange the equation \(x^{2}+y^{2}-8x+16 - 4=0\), which simplifies to \(x^{2}+y^{2}-8x + 12=0\).

Answer:

\(-8\)