Polar Coordinates: แบบฝึกหัด

A curve is defined by the polar equation \(r = 2 + 4 \cos \theta\). Determine the maximum distance from the pole to a point on this curve.

Which of the following represents the polar equation of the circle given by the Cartesian equation \(x^2 + y^2 - 4y = 0\)?

Find the exact area of the region that lies inside the circle with polar equation \(r = \sin \theta\) and outside the curve with polar equation \(r = 1 - \sin \theta\).

Which of the following Cartesian equations represents the same curve as the polar equation \( r = 4 \cos \theta \)?

For the polar curve defined by the equation \(r = 3 + 2 \sin \theta\), what is the minimum distance from the pole to a point on the curve?

Find the Cartesian equation of the curve with polar equation \(r = 6\sin\theta\).

A curve is defined by the polar equation \( r = a(1 + \cos \theta) \) for \( 0 \le \theta \le \pi \). Find the Cartesian coordinates of the point on the curve where \( \theta = \frac{\pi}{3} \), giving your answer in terms of the constant \( a \).

A region lies inside the circle with polar equation \( r = 3\cos\theta \) and outside the cardioid with polar equation \( r = 1 + \cos\theta \).
Calculate the exact area of this region.

A curve is defined by the polar equation \(r = 3 + 2\cos \theta\) for \(0 \le \theta < 2\pi\).
(a) Find the exact area of the region enclosed by the curve.
(b) Sketch the curve, clearly indicating the points where the curve intersects the initial line and the pole (if applicable).
(c) Find the polar coordinates of the point on the curve that is furthest from the pole.

The figure shows a sketch of the polar curve defined by the equation \(r = a(3 + \cos \theta)\) for \(0 \le \theta < 2\pi\), where \(a\) is a positive constant.
(a) Determine the greatest and least values of \(r\), giving your answers in terms of \(a\).
(b) By setting up and evaluating a suitable integral, find the exact area of the region enclosed by the curve. Give your answer in terms of \(a\) and \(\pi\).