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Magazine Problem 88
Find the standard form of the equation of the parabola with the given characteristics. Vertex: (-2,1)\(;\) directrix: \(x=1\)
Problem 89
Determine whether the statement is true or false. Justify your answer. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\)
Problem 89
Find the standard form of the equation of the parabola with the given characteristics. Focus: (2,2)\(;\) directrix: \(x=-2\)
Problem 90
Find the standard form of the equation of the parabola with the given characteristics. Focus: (0,0)\(;\) directrix: \(y=8\)
Problem 90
Determine whether the statement is true or false. Justify your answer. If \(\left(r, \theta_{1}\right)\) and \(\left(r, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\theta_{1}=\theta_{2}+2 \pi n\) for some integer \(n\).
Problem 91
(a) Show that the distance between the points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is \(\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}\) (b) Simplify the Distance Formula for \(\theta_{1}=\theta_{2} .\) Is the simplification what you expected? Explain. (c) Simplify the Distance Formula for \(\theta_{1}-\theta_{2}=90^{\circ}\) Is the simplification what you expected? Explain.
Problem 93
Find an equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,(4,8)$$
Problem 93
In the rectangular coordinate system, each point \((x, y)\) has a unique representation. Explain why this is not true for a point \((r, \theta)\) in the polar coordinate system.
Problem 94
Find an equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,\left(-3, \frac{2}{2}\right)$$
Problem 94
Convert the polar equation \(r=\cos \theta+3 \sin \theta\) to rectangular form and identify the graph.
