Calculus II · Power series and Taylor series
Consider the Cartesian line √(3) x + y = 6
Problem
Consider the Cartesian line \( \sqrt{3}\, x + y = 6 \). (a) Convert the equation to polar form and solve for \( r \) as a function of \(\theta\) on the set of \(\theta\) for which \( r>0 \). (b) Rewrite that polar equation in the form \( r = p\sec(\theta-\alpha) \) for an explicit constant \( p>0 \) and an explicit angle \(\alpha\in(-\pi,\pi]\). (c) Find the polar coordinates \((r,\theta)\) with \( r>0 \) of the point on the line nearest the pole, and give the corresponding Cartesian coordinates. (d) Convert the Cartesian point \( (3,\, 3\sqrt{3}) \) to polar coordinates with \( r>0 \) and \(\theta\in(-\pi,\pi]\), and determine whether this point lies on the line.
Hint
Substitute \(x=r\cos\theta\) and \(y=r\sin\theta\), then factor \(r\). The linear combination of cosine and sine is a single shifted cosine.
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