Chapter 10

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$r=\frac{3}{1+\sin \theta}$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

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 \(t.\) \(x=3-5 t, y=4+2 t ; t=1\)

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. $$x y=-1 ; \theta=45^{\circ}$$

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$r=\frac{3}{1+\cos \theta}$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 $$

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 \(t.\) \(x=7-4 t, y=5+6 t ; t=1\)

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. $$x y=-4 ; \theta=45^{\circ}$$

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

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{9}+\frac{y^{2}}{36}=1 $$

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 \(t.\) \(x=t^{2}+1, y=5-t^{3} ; t=2\)

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 polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{6}{3+2 \cos \theta} $$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{16}+\frac{y^{2}}{49}=1 $$

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 \(t.\) \(x=t^{2}+3, y=6-t^{3} ; t=2\)

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. $$13 x^{2}-10 x y+13 y^{2}-72=0 ; \theta=45^{\circ}$$

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

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{25}+\frac{y^{2}}{64}=1 $$

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 \(t.\) \(x=4+2 \cos t, y=3+5 \sin t ; t=\frac{\pi}{2}\)

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. $$23 x^{2}+26 \sqrt{3} x y-3 y^{2}-144=0 ; \theta=30^{\circ}$$

In Exercises 5–16, 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. $$ \text { Foci: }(0,-3),(0,3) ; \text { vertices: }(0,-1),(0,1) $$

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

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{49}+\frac{y^{2}}{36}=1 $$

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 \(t.\) \(x=2+3 \cos t, y=4+2 \sin t ; t=\pi\)

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. $$13 x^{2}-6 \sqrt{3} x y+7 y^{2}-16=0 ; \theta=60^{\circ}$$

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ y^{2}=4 x $$

find the standard form of the equation of each hyperbola satisfying the given conditions. $$ \text { Foci: }(0,-6),(0,6) ; \text { vertices: }(0,-2),(0,2) $$

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{12}{2-4 \cos \theta} $$

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 \(t.\) \(x=\left(60 \cos 30^{\circ}\right) t, y=5+\left(60 \sin 30^{\circ}\right) t-16 t^{2} ; t=2\)

Write the appropriate rotation formulas so that in a rotated system the equation has no \(x^{\prime} y^{\prime}\) -term. $$x^{2}+x y+y^{2}-10=$$

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ y^{2}=-8 x $$

find the standard form of the equation of each hyperbola satisfying the given conditions. $$ \text { Foci: }(-4,0),(4,0) ; \text { vertices: }(-3,0),(3,0) $$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{49}+\frac{y^{2}}{81}=1 $$

a. Identify the conic section that each polarequation represents. b. Describe the location of a directrix from the focus located at the pole. $$ r=\frac{12}{2+4 \cos \theta} $$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{64}+\frac{y^{2}}{100}=1 $$

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 \(t.\) \(x=\left(80 \cos 45^{\circ}\right) t, y=6+\left(80 \sin 45^{\circ}\right) t-16 t^{2} ; t=2\)

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ y^{2}=-12 x $$

find the standard form of the equation of each hyperbola satisfying the given conditions. $$\text { Foci: }(-7,0),(7,0) ; \text { vertices: }(-5,0),(5,0)$$

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{\frac{2}{4}}+\frac{y^{2}}{\frac{25}{4}}=1 $$

\(x=t+2, y=t^{2} ;-2 \leq t \leq 2\)

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ x^{2}=12 y $$

find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: \((0,-6),(0,6) ;\) asymptote: \(y=2 x\)

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ \frac{x^{2}}{\frac{81}{4}}+\frac{y^{2}}{\frac{25}{16}}=1 $$

Use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \( t.\) \(x=t-1, y=t^{2} ;-2 \leq t \leq 2\)

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ x^{2}=8 y $$

find the standard foTherefore, the standard form of the equation of the parabola that satisfies the given conditions is \(\frac{x^{2}}{16}-\frac{y^{2}}{64}=1\)rm of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: \((-4,0),(4,0) ;\) asymptote: \(y=2 x\)

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ x^{2}=1-4 y^{2} $$

Use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \( t.\) \(x=t-2, y=2 t+1 ;-2 \leq t \leq 3\)

In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$ x^{2}=-16 y $$