Chapter 8

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 \((0,-2),\) and point on graph (5,0) .

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

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

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

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

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

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

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

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

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: \(x=-5 ; \quad e=\frac{3}{4}\)

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

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

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 \(\theta\). \(x y=2\)

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.

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

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 \(\theta\). \(x^{2}+x y+y^{2}=4\)

Given \(2 x^{2}+k x y+12 y^{2}+10 x-16 y+28=0\) find \(k\) for the graph to be an ellipse.

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

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. \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\)

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

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 \(\theta\). \(2 x^{2}+4 x y+2 y^{2}=9\)

Given \(3 x^{2}+k x y+4 y^{2}-6 x+20 y+128=0\) find \(k\) for the graph to be a hyperbola.

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. \(V(1,2),\) Endpoints (-5,5),(7,5)

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. \(\frac{(x-2)^{2}}{16}-\frac{(y+3)^{2}}{25}=1\)

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\). \(\frac{(x+6)^{2}}{16}+\frac{(y-6)^{2}}{36}=1\)

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 \(\theta\). \(16 x^{2}+24 x y+9 y^{2}=4\)

Given \(3 x^{2}+k x y+4 y^{2}-6 x+20 y+128=0\) find \(k\) for the graph to be a hyperbola.

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. \(V(-3,-1),\) Endpoints (0,5) , (0,-7)

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. \(-4 x^{2}-16 x+y^{2}-2 y-19=0\)

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\). \(\frac{(x+1)^{2}}{4}+\frac{(y-2)^{2}}{5}=1\)

Given \(6 x^{2}+12 x y+k y^{2}+16 x+10 y+4=0\) find \(k\) for the graph to be an ellipse.

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. \(V(4,-3)\), Endpoints \(\left(5,-\frac{7}{2}\right)\), \(\left(3,-\frac{7}{2}\right)\)

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. \(4 x^{2}-24 x-y^{2}-4 y+16=0\)

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\). \(4 x^{2}-8 x+9 y^{2}-72 y+112=0\)

The mirror in an automobile headlight has a parabolic crosssection with the light bulb at the focus. On a schematic, the equation of the parabola is given as \(x^{2}=4 y\). At what coordinates should you place the light bulb?

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes \(y=x \quad\) and \(y=-x,\) and its closest distance to the center fountain is 5 yards.

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes \(y=2 x\) and \(y=-2 x,\) and its closest distance to the center fountain is 6 yards.

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?