91Ó°ÊÓ

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. $$\text { Left half of } \frac{x^{2}}{36}+\frac{y^{2}}{25}=1$$

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

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r^{2}\left(4 \sin ^{2} \theta-9 \cos ^{2} \theta\right)=36$$

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$\left\\{\begin{aligned}y^{2}-4 x^{2} &=16 \\\y-x &=4\end{aligned}\right.$$

Find an equation for the indicated half of the parabola. Right half of \((x-4)^{2}=y-5\)

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r^{2}\left(\cos ^{2} \theta+4 \sin ^{2} \theta\right)=16$$

Exer. \(47-50\) : Find an equation for the indicated half of the ellipse. $$\text { Right half of } \frac{x^{2}}{9}+\frac{y^{2}}{121}=1$$

Lissajous figures are used in the study of electrical circuits to determine the phase difference \(\phi\) between a known voltage \(V_{1}(t)=A \sin (\omega t)\) and an unknown voltage \(V_{2}(\vec{t})=B \sin (\omega t+\phi)\) having the same frequency. The voltages are graphed parametrically as \(x=V_{1}(t)\) and \(y=V_{2}(t)\) If \(\phi\) is acute, then $$\phi=\sin ^{-1} \frac{y_{\mathrm{int}}}{y_{\max }}$$ 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. $$\begin{aligned}&V_{1}(t)=6 \sin (120 \pi t), \quad V_{2}(t)=5 \cos (120 \pi t)&0 \leq t \leq 0.02\end{aligned}$$

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$\left\\{\begin{array}{l}x^{2}-y^{2}=4 \\\y^{2}-3 x=0\end{array}\right.$$

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. $$y=\sqrt{x-6}-2$$