Chapter 12

Suppose we want to graph an equation in \(x\) and \(y .\) (a) If we replace \(x\) by \(x-3,\) the graph of the equation is shifted to the _____ by 3 units. If we replace \(x\) by \(x+3,\) the graph of the equation is shifted to the ____ by 3 units. (b) If we replace \(y\) by \(y-1,\) the graph of the equation is shifted _____ by 1 unit. If we replace \(y\) by \(y+1,\) the graph of the equation is shifted _____ by 1 unit.

An ellipse is the set of all points in the plane for which the ________ of the distances from two fixed points \(F_{1}\) and \(F_{2}\) is constant. The points \(F_{1}\) and \(F_{2}\) are called the __________ of the ellipse.

A parabola is the set of all points in the plane that are equidistant from a fixed point called the _____ and a fixed line called the _____ of the parabola.

The graphs of \(x^{2}=12 y\) and \((x-3)^{2}=12(y-1)\) are given. Label the focus, directrix, and vertex on each parabola. (GRAPH NOT COPY)

The graph of the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (_, _) and (_, _) and foci \(( \pm c, 0)\) where \(c=\) _________ So the graph of \(\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1\) is an ellipse with vertices (_, _) and (_, _) and foci (_, _) and (_, _)

The graph of the equation \(x^{2}=4 p y\) is a parabola with focus F(_____,_____) and directrix \(y=\) _____. So the graph of \(x^{2}=12 y\) is a parabola with focus \(F\) (_____,_____) and directrix \(y=\) _____.

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{2}{3},\) directrix \(x=3\)

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (1,1), \quad \phi=45^{\circ} $$

The graph of the equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (_, _) and (_, _) and foci \((0, \pm c),\) where \(c=\) __________ \((0, \pm c),\) where \(c=\) is an ellipse with vertices (_, _) and (_, _) and foci (_,_) and (_, _)

The graph of the equation \(y^{2}=4 p x\) is a parabola with focus F (_____,_____) and directrix \(x=\) _____ So the graph of \(y^{2}=12 x\) is a parabola with focus \(F\) (_____,_____) and directrix \(x=\) _____.

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(\frac{4}{3},\) directrix \(x=-3\)

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (-2,1), \quad \phi=30^{\circ} $$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, directrix \(y=2\)

\(5-8\) . Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$ \frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{4}=1 $$

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (3,-\sqrt{3}), \quad \phi=60^{\circ} $$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\frac{1}{2},\) directrix \(y=-4\)

\(5-8\) . Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$ \frac{(x-3)^{2}}{16}+(y+3)^{2}=1 $$

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (2,0), \quad \phi=15^{\circ} $$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(4,\) directrix \(r=5 \sec \theta\)

\(5-8\) . Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$ \frac{x^{2}}{9}+\frac{(y+5)^{2}}{25}=1 $$

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (0,2), \quad \phi=55^{\circ} $$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6,\) directrix \(r=2 \csc \theta\)

\(5-8\) . Find the center, foci, and vertices of the ellipse, and determine the lengths of the major and minor axes. Then sketch the graph. $$ \frac{(x+2)^{2}}{4}+y^{2}=1 $$

\(3-8\) . Determine the \(X Y\) -coordinates of the given point if the coordinate axes are rotated through the indicated angle. $$ (\sqrt{2}, 4 \sqrt{2}), \quad \phi=45^{\circ} $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ \frac{x^{2}}{4}-\frac{y^{2}}{16}=1 $$

\(9-12\) . Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$ (x-3)^{2}=8(y+1) $$

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x^{2}-3 y^{2}=4, \quad \phi=60^{\circ} $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ \frac{y^{2}}{9}-\frac{x^{2}}{16}=1 $$

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity 0.4, vertex at \((2,0)\)

\(9-12\) . Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$ (y+5)^{2}=-6 x+12 $$

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ y=(x-1)^{2}, \quad \phi=45^{\circ} $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ \frac{x^{2}}{16}+\frac{y^{2}}{25}=1 $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ y^{2}-\frac{x^{2}}{25}=1 $$

\(9-12\) . Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$ -4\left(x+\frac{1}{2}\right)^{2}=y $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 9 x^{2}+4 y^{2}=36 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ x^{2}=9 y $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ \frac{x^{2}}{2}-y^{2}=1 $$

\(9-12\) . Find the vertex, focus, and directrix of the parabola. Then sketch the graph. $$ y^{2}=16 x-8 $$

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x^{2}+2 y^{2}=16, \quad \phi=\sin ^{-1} \frac{3}{5} $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ 4 x^{2}+25 y^{2}=100 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ x^{2}=y $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ x^{2}-y^{2}=1 $$

\(13-16\) . Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$ \frac{(x+1)^{2}}{9}-\frac{(y-3)^{2}}{16}=1 $$

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x^{2}+2 \sqrt{3} x y-y^{2}=4, \quad \phi=30^{\circ} $$

Find the vertices, foci, and eccentricity of the ellipse. Determine the lengths of the major and minor axes, and sketch the graph. $$ x^{2}+4 y^{2}=16 $$

\(11-22\) . Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$ y^{2}=4 x $$

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$ 9 x^{2}-4 y^{2}=36 $$

\(13-16\) . Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$ (x-8)^{2}-(y+6)^{2}=1 $$

\(9-14\) . Determine the equation of the given conic in \(X Y\) -coordinates when the coordinate axes are rotated through the indicated angle. $$ x y=x+y, \quad \phi=\pi / 4 $$