Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x²+y²-2x+12y-12=0

검증된 단계별 안내

Start by rewriting the given equation: \(x^2 + y^2 - 2x + 12y - 12 = 0\).

Group the \(x\) terms and \(y\) terms together: \((x^2 - 2x) + (y^2 + 12y) = 12\) (move the constant to the right side).

Complete the square for the \(x\) terms: take half of \(-2\), which is \(-1\), and square it to get \(1\). Add \(1\) inside the \(x\) group.

Complete the square for the \(y\) terms: take half of \(12\), which is \(6\), and square it to get \(36\). Add \(36\) inside the \(y\) group.

Since you added \(1\) and \(36\) to the left side, add \(1 + 36 = 37\) to the right side as well to keep the equation balanced. Then rewrite the equation as \((x - 1)^2 + (y + 6)^2 = 12 + 37\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Standard Form of a Circle Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's properties.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions as perfect square trinomials. This technique is essential to transform the given equation into the standard form of a circle.

Determining the Nature of the Graph

After rewriting the equation, the value of r^2 determines the graph's nature: if r^2 > 0, it's a circle; if r^2 = 0, it's a single point; if r^2 < 0, the graph does not exist in the real plane.

Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x2+y2-2x+12y-12=0