91Ó°ÊÓ

Find an equation for the hyperbola that has its center at the origin and satisfled the given conditions. Vertices \(V(\pm 4,0),\) passing through \((8,2)\)

Find an equation for the set of points in an xy-plane that are equidistant from the point \(P\) and the line \(l\). $$P(-6,3) ; \quad l: x=-2$$

Find an equation for the indicated half of the parabola. Lower half of \((y+1)^{2}=x+3\)

Lissajous figures are used in the study of electrical circults to determine the phase difference \(\phi\) between a known voltage \(V_{1}(t)=A \sin (\omega t)\) and an unknown voltage \(V_{2}(t)=B \sin (\omega t+\phi)\) having the same frequency. The voltages are graphed parametrically as \(x=V_{1}(t)\) and \(y=V_{2}(t)\) If \(\phi\) is acute, then $$\phi=\sin ^{-1} \frac{y_\text{int}}{y_{\max }}$$ where \(y_{\text {int }}\) is the nonnegative \(y\) -intercept and \(y_{\max }\) is the maximum \(y\) -value on the curve. (a) Graph the parametric curve \(x=V_{1}(t)\) and \(y=V_{2}(t)\) for the specified range of \(t\) (b) Use the graph to approximate \(\phi\) in degrees. \(V_{1}(t)=6 \sin (120 \pi t), \quad V_{2}(t)=5 \cos (120 \pi t)\) \(-0 \leq t \leq 0.02\)

A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the receiver be placed to receive the greatest intensity of sound waves?

Determine whether the graph of the equation Is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$y=-1+\sqrt{1-\frac{(x-3)^{2}}{16}}$$

A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the width of the dish if the depth is to be 2 feet.

Mercury's orbit The planet Mercury travels in an elliptical orbit that has eccentricity 0.206 and major axis of length 0.774 AU. Find the maximum and minimum distances between Mercury and the sun.