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+4x+14y=-54

검증된 단계별 안내

Start by rewriting the given equation: \(x^2 + y^2 + 4x + 14y = -54\).

Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 + 14y) = -54\).

Complete the square for the \(x\) terms: take half of 4, which is 2, and square it to get 4. Add and subtract 4 inside the equation.

Complete the square for the \(y\) terms: take half of 14, which is 7, and square it to get 49. Add and subtract 49 inside the equation.

Rewrite the equation as perfect square trinomials: \((x + 2)^2 - 4 + (y + 7)^2 - 49 = -54\). Then, combine constants on the right side to find the standard form of the circle equation.

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주요 개념

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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.

Interpreting the Radius and Existence of the Circle

After rewriting the equation, the radius squared (r^2) must be positive for a circle to exist. If r^2 = 0, the graph is a single point; if r^2 is negative, 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+4x+14y=-54