91Ó°ÊÓ

Graph each ellipse and locate the foci. $$\frac{x^{2}}{9}+\frac{y^{2}}{36}=1$$

Write each equation in terms of a rotated \(x^{\prime} y^{\prime}-\) system using \(\theta,\) the angle of rotation. Write the equation involving \(x^{\prime}\) and \(y^{\prime}\) in standard form. $$x^{2}-4 x y+y^{2}-3=0 ; \theta=45^{\circ}$$

a. Identify the conic section that each polar equation represents. b. Describe the location of a directrix from the focus located at the pole. $$r=\frac{6}{3-2 \cos \theta}$$

Graph each ellipse and locate the foci. $$\frac{x^{2}}{16}+\frac{y^{2}}{49}=1$$

Write each equation in terms of a rotated \(x^{\prime} y^{\prime}-\) system using \(\theta,\) the angle of rotation. Write the equation involving \(x^{\prime}\) and \(y^{\prime}\) in standard form. $$13 x^{2}-10 x y+13 y^{2}-72=0 ; \theta=45^{\circ}$$

In Exercises \(1-8,\) parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of \(t\). $$x=t^{2}+3, y=6-t^{3} ; t=2$$

a. Identify the conic section that each polar equation represents. b. Describe the location of a directrix from the focus located at the pole. $$r=\frac{6}{3+2 \cos \theta}$$

a. Identify the conic section that each polar equation represents. b. Describe the location of a directrix from the focus located at the pole. $$r=\frac{8}{2+2 \sin \theta}$$

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$y^{2}=16 x$$

Find the standard form of the equation of each hyperbola satisfying the given conditions. $$\text { Foci: }(0,-3),(0,3) ; \text { vertices: }(0,-1),(0,1)$$