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Chapter 6: Problem 30

Eliminate the parameter and obtain the standard form of the rectangular equation. Circle: \(x=h+r \cos \theta, y=k+r \sin \theta\)

Short Answer

Expert verified
The standard form of the given parametric equations of the circle is \((x - h)^2 + (y - k)^2 = r^2\)

Step by step solution

01

Identify the Parameterized Equations

Given are two parameterized equations of a circle: \(x = h + r \cos \theta\) and \(y = k + r \sin \theta\). Here, \(r\) represents the radius of the circle, while \(h\) and \(k\) represent the coordinates of the circle's center.
02

Isolate Cosine and Sine Terms

Rearrange both equations to isolate the cosine and sine terms on one side. This can be achieved by subtracting \(h\) from the \(x\)-equation and \(k\) from the \(y\)-equation: \(\cos \theta = (x - h)/r\) and \(\sin \theta = (y - k)/r\)
03

Apply Pythagorean Identity

To eliminate the parameter \( \theta \), square both equations to get \(\cos^2 \theta = (x - h)^2/r^2\) and \(\sin^2 \theta = (y - k)^2/r^2\), and then add them together. The Pythagorean identity for sine and cosine states that \(\sin^2 \theta + \cos^2 \theta = 1\). \nApplying this in the context of the two equations, it results in \((x - h)^2/r^2 + (y - k)^2/r^2 = 1\)
04

Obtain the Standard Form

To simplify, multiply the entire equation by \(r^2\) to clear the denominator, thus yielding the standard form of a circle: \((x - h)^2 + (y - k)^2 = r^2\)

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Most popular questions from this chapter

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Vertical axis and passes through the point (-3,-3)

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(\frac{x^{2}}{9}+\frac{y^{2}}{9}=1\)

Explain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) \(y=a(x-h)^{2}+k, \quad a \neq 0\) (b) \((x-h)^{2}=4 p(y-k), \quad p \neq 0\) (c) \((y-k)^{2}=4 p(x-h), \quad p \neq 0\)

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(\frac{(x-3)^{2}}{25 / 4}+\frac{(y-1)^{2}}{25 / 4}=1\)
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