The parametric equation of the circle x^2+y^2+p x+p y=0 is (a) x=(-p)/(2), y=(-p)/(2) (b) x=(p)/

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The parametric equation of the circle x^2+y^2+p x+p y=0 is...

The parametric equation of the circle $x^{2}+y^{2}+p x+p y=0$ is (a) $\mathrm{x}=\frac{-\mathrm{p}}{2}, \mathrm{y}=\frac{-\mathrm{p}}{2}$ (b) $\mathrm{x}=\frac{\mathrm{p}}{2} \cos \theta, \mathrm{y}=\frac{\mathrm{p}}{2} \sin \theta$ (c) $x=\frac{-p}{2}+\frac{p}{\sqrt{2}} \cos \theta, y=\frac{-p}{2}+\frac{p}{\sqrt{2}} \sin \theta$ (d) $\mathrm{x}=\frac{\mathrm{p}}{\sqrt{2}} \cos \theta, \mathrm{y}=\frac{\mathrm{p}}{\sqrt{2}} \sin \theta$

The parametric equation of the circle $x^{2}+y^{2}+p x+p y=0$ is
(a) $\mathrm{x}=\frac{-\mathrm{p}}{2}, \mathrm{y}=\frac{-\mathrm{p}}{2}$
(b) $\mathrm{x}=\frac{\mathrm{p}}{2} \cos \theta, \mathrm{y}=\frac{\mathrm{p}}{2} \sin \theta$
(c) $x=\frac{-p}{2}+\frac{p}{\sqrt{2}} \cos \theta, y=\frac{-p}{2}+\frac{p}{\sqrt{2}} \sin \theta$
(d) $\mathrm{x}=\frac{\mathrm{p}}{\sqrt{2}} \cos \theta, \mathrm{y}=\frac{\mathrm{p}}{\sqrt{2}} \sin \theta$

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a perfect square form. This method is particularly useful when dealing with conic sections because it allows one to rewrite quadratic equations in a form that easily reveals properties such as the center and radius of a circle.

Standard Equation of a Circle

The standard equation of a circle in the Cartesian plane is written as (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. Recognizing and converting a given quadratic equation into this standard form is essential for identifying the circle's geometric properties.

Parametric Representation

A parametric representation of a curve describes the coordinates of points on the curve in terms of a parameter, often denoted by ? or t. For a circle, this is typically expressed as x = h + r cos ? and y = k + r sin ?, which traces the entire circle as the angle ? varies from 0 to 2?.

Identification of Center and Radius

Extracting the center and radius from a quadratic equation involves rewriting the equation, often using the method of completing the square, to match the standard form of a circle. Once in standard form, the values of h, k, and r can be directly read off, providing crucial information for further analysis and graphical representation.

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