Question

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

Short answer

The standard equation of the circle in the rectangular coordinate system is \( (x - h)^2 + (y - k)^2 = r^2 \).

Step-by-step solution

Express \( \cos \theta \) and \( \sin \theta \) in terms of \( x \) and \( y \)

Rearrange the given equations to express \( \cos \theta \) and \( \sin \theta \) in terms of \( x \) and \( y \) respectively.\n From \( x = h + r \cos \theta \), we get \( \cos \theta = \frac{x - h}{r} \). \n Similarly, from \( y = k + r \sin \theta \), we get \( \sin \theta = \frac{y - k}{r} \).

Use the Pythagorean identity

Substitute \( \cos \theta = \frac{x - h}{r} \) and \( \sin \theta = \frac{y - k}{r} \) into the Pythagorean trigonometric identity, \( \sin^2 \theta + \cos^2 \theta = 1 \).\n This gives: \( \left(\frac{x - h}{r}\right)^2 + \left(\frac{y - k}{r}\right)^2 = 1 \).

Convert to standard form

The expression obtained in Step 2 is very close to the standard form. To get the exact standard form of the equation of a circle, we simply have to clear the denominator \( r^2 \) in the left hand side:\n This results in: \( (x - h)^2 + (y - k)^2 = r^2 \).\n This is the standard form of the equation of a circle in the rectangular coordinate system, where \( (h, k) \) are the coordinates of the center of the circle and \( r \) is the radius.