91Ó°ÊÓ

Give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation in Cartesian coordinates. Eccentricity: 1.25 Foci: \(\quad(0,\pm 5)\)

Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1, \quad x=2$$

Find parametric equations for the semicircle $$x^{2}+y^{2}=a^{2}, \quad y>0$$ using as parameter the slope \(t=d y / d x\) of the tangent to the curve at \((x, y)\)

If is continuous, the average value of the polar coordinate rover the curve \(r=f(\theta), \alpha \leq \theta \leq \beta,\) with respect to \(\theta\) is given by the formula $$r_{\mathrm{av}}=\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta} f(\theta) d \theta$$ Use this formula to find the average value of \(r\) with respect to \(\theta\) over the following curves \((a>0)\) a. The cardioid \(r=a(1-\cos \theta)\) b. The circle \(r=a\) c. The circle \(r=a \cos \theta, \quad-\pi / 2 \leq \theta \leq \pi / 2\)

Graph the nephroid of Freeth: $$r=1+2 \sin \frac{\theta}{2}$$

A wheel of radius \(a\) rolls along a horizontal straight line without slipping. Find parametric equations for the curve traced out by a point \(P\) on a spoke of the wheel \(b\) units from its center. As parameter, use the angle \(\theta\) through which the wheel turns. The curve is called a trochoid, which is a cycloid when \(b=a\)

Volume Find the volume swept out by revolving the region bounded by the \(x\) -axis and one arch of the cycloid $$ x=t-\sin t, \quad y=1-\cos t $$ about the \(x\) -axis.

Give polar coordinates for their centers and identify their radii. $$r=4 \cos \theta$$

Give polar coordinates for their centers and identify their radii. $$r=6 \sin \theta$$

Replace the Cartesian equations with equivalent polar equations. $$x-y=3$$