Question

Sketch the graph of the equation. $$x^{2}+y^{2}+6 x-12 y+45=0$$

   Sketch the graph of the equation.
$$x^{2}+y^{2}+6 x-12 y+45=0$$

College Algebra

College Algebra

James Stewart,… 7th Edition

Chapter 1, Problem 93

Instant Answer

Solved by Expert Ankit Gupta

07/24/2020

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The given equation is $x^{2}+y^{2}+6 x-12 y+45=0$. Show more…

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Sketch the graph of the equation. $$x^{2}+y^{2}+6 x-12 y+45=0$$

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Key Concepts

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into a perfect square trinomial. In the context of circle equations, it involves reordering and regrouping the x and y terms and then adding and subtracting the appropriate constants to form complete squares. This technique facilitates rewriting the equation into a form that clearly shows the geometric properties of the circle.

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius. Converting a general quadratic equation into this form allows one to directly read off the center and the radius, which are key for understanding and graphing the circle.

Graphing a Circle

Graphing a circle involves plotting its center and using the radius to determine the distance from the center to any point on the circumference. This process provides a clear visual representation of the circle in the coordinate plane, demonstrating its size and location. Recognizing the center and radius from the standard form is essential for accurate drawing.

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