Chapter 8

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) 'term. $$16 x^{2}+24 x y+9 y^{2}-60 x+80 y=0$$

Given information about the graph of the hyperbola, find its equation. Vertices at \((1,1)\) and \((11,1)\) and one focus at \((12,1)\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-1 ; e=1\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-1 ; e=1\)

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (-2,3)\(;\) directrix is \(x=-\frac{7}{2},\) focus is \(\left(-\frac{1}{2}, 3\right)\).

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center at the origin, symmetric with respect to the \(x\) - and \(y\) -axes, focus at \((3,0),\) and major axis is twice as long as minor axis.

For the following exercises, given information about the graph of the hyperbola, find its equation. Center: (0,0) ; vertex: (0,-13) ; one focus: \((0, \sqrt{313})\).

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) 'term. $$13 x^{2}-6 \sqrt{3} x y+7 y^{2}-16=0$$

Given information about the graph of the hyperbola, find its equation. Center: \((0,0) ;\) vertex: \((0,-13) ;\) one focus: \((0, \sqrt{313})\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-\frac{1}{4} ; e=\frac{7}{2}\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-\frac{1}{4} ; e=\frac{7}{2}\)

For the following exercises, find the equation of the parabola given information about its graph. Vertex is \((\sqrt{2},-\sqrt{3})\); directrix is \(x=2 \sqrt{2}\), focus is \((0,-\sqrt{3})\)

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center \((4,2) ;\) vertex \((9,2) ;\) one focus: \((4+2 \sqrt{6}, 2)\)

For the following exercises, given information about the graph of the hyperbola, find its equation. Center: (4,2)\(;\) vertex: (9,2)\(;\) one focus: \((4+\sqrt{26}, 2)\)

For the following exercises, graph the equation relative to the \(x^{\prime} y^{\prime}\) system in which the equation has no \(x^{\prime} y^{\prime}\) 'term. $$4 x^{2}-4 x y+y^{2}-8 \sqrt{5} x-16 \sqrt{5} y=0$$

Given information about the graph of the hyperbola, find its equation. Center: \((4,2) ;\) vertex: \((9,2) ;\) one focus: \((4+\sqrt{26}, 2)\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=\frac{2}{5} ; e=\frac{7}{2}\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=\frac{2}{5} ; e=\frac{7}{2}\)

For the following exercises, find the equation of the parabola given information about its graph. Vertex is (1,2)\(;\) directrix is \(y=\frac{11}{3},\) focus is \(\left(1, \frac{1}{3}\right)\).

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center \((3,5) ;\) vertex \((3,11) ;\) one focus: \((3,5+4 \sqrt{2})\)

For the following exercises, given information about the graph of the hyperbola, find its equation. Center: (3,5)\(;\) vertex: (3,11)\(;\) one focus: \((3,5+2 \sqrt{10})\)

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$6 x^{2}-5 \sqrt{3} x y+y^{2}+10 x-12 y=0$$

Given information about the graph of the hyperbola, find its equation. Center: \((3,5) ;\) vertex: \((3,11) ;\) one focus: \((3,5+2 \sqrt{10})\)

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=4 ; e=\frac{3}{2}\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=4 ; e=\frac{3}{2}\)

For the following exercises, use the given information about the graph of each ellipse to determine its equation. Center \((-3,4) ;\) vertex \((1,4) ;\) one focus: \((-3+2 \sqrt{3}, 4)\)

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$6 x^{2}-5 x y+6 y^{2}+20 x-y=0$$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-2 ; e=\frac{8}{3}\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-2 ; e=\frac{8}{3}\)

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$6 x^{2}-8 \sqrt{3} x y+14 y^{2}+10 x-3 y=0$$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-5 ; e=\frac{3}{4}\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-5 ; e=\frac{3}{4}\)

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$4 x^{2}+6 \sqrt{3} x y+10 y^{2}+20 x-40 y=0$$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=2 ; e=2.5\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=2 ; e=2.5\)

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$8 x^{2}+3 x y+4 y^{2}+2 x-4=0$$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-3 ; e=\frac{1}{3}\)

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=-3 ; e=\frac{1}{3}\)

For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. $$16 x^{2}+24 x y+9 y^{2}+20 x-44 y=0$$

Recall from Rotation of Axes that equations of conics with an \(x y\) term have rotated graphs. For the following exercises, express each equation in polar form with \(r\) as a function of \(\theta\). $$ x y=2 $$

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ x y=2 $$

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. $$ V(0,0), \text { Endpoints }(2,1),(-2,1) $$

For the following exercises, express the equation for the hyperbola as two functions, with \(y\) as a function of \(x .\) Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. $$ \frac{x^{2}}{4}-\frac{y^{2}}{9}=1 $$

For the following exercises, determine the value of \(k\) based on the given equation. Given \(4 x^{2}+k x y+16 y^{2}+8 x+24 y-48=0\) find \(k\) for the graph to be a parabola.

Express the equation for the hyperbola as two functions, with y as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. \(\frac{x^{2}}{4}-\frac{y^{2}}{9}=1\)

Recall from Rotation of Axes that equations of conics with an \(x y\) term have rotated graphs. For the following exercises, express each equation in polar form with \(r\) as a function of \(\theta\). $$ x^{2}+x y+y^{2}=4 $$

Express each equation in polar form with \(r\) as a function of \(\theta\). $$ x^{2}+x y+y^{2}=4 $$

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. $$ V(0,0), \text { Endpoints }(-2,4),(-2,-4) $$

For the following exercises, find the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\) $$ \frac{(x-3)^{2}}{9}+\frac{(y-3)^{2}}{16}=1 $$

For the following exercises, express the equation for the hyperbola as two functions, with \(y\) as a function of \(x .\) Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. $$ \frac{y^{2}}{9}-\frac{x^{2}}{1}=1 $$