Multiple Choice
Determine if the equation is a circle, and if it is, find its center and radius.
8. Conic Sections
Circles
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Determine if the equation is a circle, and if it is, find its center and radius.
A
Is a circle, center = , radius .
B
Is a circle, center = , radius .
C
Is a circle, center = , radius .
D
Is not a circle.
Verified step by step guidance1
Start by rewriting the given equation: \( x^2 + y^2 - 2x + 4y - 4 = 0 \).
To determine if this is a circle, we need to rewrite it in the standard form of a circle's equation: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
Complete the square for the \(x\) terms: \( x^2 - 2x \). Take half of the coefficient of \(x\), square it, and add it inside the equation: \( (x - 1)^2 - 1 \).
Complete the square for the \(y\) terms: \( y^2 + 4y \). Take half of the coefficient of \(y\), square it, and add it inside the equation: \( (y + 2)^2 - 4 \).
Rewrite the equation incorporating the completed squares: \( (x - 1)^2 + (y + 2)^2 = 9 \). This shows the equation is a circle with center \((1, -2)\) and radius \(3\).
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