Question

the equation for a circle is shown below. (x + 6)^2 + (y - 7)^2 = 64 what is the expanded form of this standard equation? 2 of 4 questio x^2 + y^2 + 12x - 14y - 64 = 0 x^2 + y^2 + 12x - 14y + 21 = 0 x^2 + y^2 + 13 = 0 x^2 + y^2 + 12x - 14y + 85 = 0

Explanation:

Step1: Expand \((x + 6)^2\)

Using the formula \((a+b)^2=a^{2}+2ab + b^{2}\), for \(a = x\) and \(b = 6\), we have \((x + 6)^2=x^{2}+12x + 36\).

Step2: Expand \((y - 7)^2\)

Using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), for \(a=y\) and \(b = 7\), we get \((y - 7)^2=y^{2}-14y + 49\).

Step3: Substitute expansions into the original equation

The original equation \((x + 6)^2+(y - 7)^2=64\) becomes \(x^{2}+12x + 36+y^{2}-14y + 49=64\).

Step4: Simplify the equation

Combine like - terms: \(x^{2}+y^{2}+12x-14y+(36 + 49-64)=0\).
Calculate \(36 + 49-64=21\).

Answer:

\(x^{2}+y^{2}+12x-14y + 21=0\) (the second option)