Chapter 7

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{x^{2}}{16}-\frac{y^{2}}{25}=1 $$

In Problems 1-20, find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{x^{2}}{25}+\frac{y^{2}}{9}=1 $$

Graph the given point. $$ (1,1,5) $$

In Problems \(1-24,\) find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(y^{2}=4 x\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{x^{2}}{16}-\frac{y^{2}}{25}=1 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

Graph the given point. $$ (0,0,4) $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(y^{2}=\frac{7}{2} x\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{y^{2}}{64}-\frac{x^{2}}{9}=1 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

Graph the given point. $$ (3,4,0) $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(y^{2}=-\frac{4}{3} x\)

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{x^{2}}{4}+\frac{y^{2}}{10}=1 $$

Graph the given point. $$ (6,0,0) $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(y^{2}=-10 x\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ 4 x^{2}-16 y^{2}=64 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ 9 x^{2}+16 y^{2}=144 $$

Graph the given point. $$ (6,-2,0) $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(x^{2}=-16 y\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ 5 x^{2}-5 y^{2}=25 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ 2 x^{2}+y^{2}=4 $$

Graph the given point. $$ (5,-4,3) $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(x^{2}=\frac{1}{10} y\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ y^{2}-5 x^{2}=20 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ 9 x^{2}+4 y^{2}=36 $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(x^{2}=28 y\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ 9 x^{2}-16 y^{2}+144=0 $$

Describe geometrically all points \(P(x, y, z)\) whose coordinates satisfy the given conditions. $$ z=5 $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(x^{2}=-64 y\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{(x-5)^{2}}{4}-\frac{(y+1)^{2}}{49}=1 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{(x-1)^{2}}{49}+\frac{(y-3)^{2}}{36}=1 $$

Describe geometrically all points \(P(x, y, z)\) whose coordinates satisfy the given conditions. $$ x=2, y=3 $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \((y-1)^{2}=16 x\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{(x+2)^{2}}{10}-\frac{(y+4)^{2}}{25}=1 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{(x+1)^{2}}{25}+\frac{(y-2)^{2}}{36}=1 $$

Describe geometrically all points \(P(x, y, z)\) whose coordinates satisfy the given conditions. $$ x=4, y=-1, z=7 $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \((y+3)^{2}=-8(x+2)\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{(y-4)^{2}}{36}-x^{2}=1 $$

In Problems \(11-16,\) use rotation of axes to eliminate the \(x y\) -term in the given equation. Identify the conic and graph. $$ x^{2}+x y+y^{2}=4 $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \((x+5)^{2}=-4(y+1)\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ \frac{\left(y-\frac{1}{4}\right)^{2}}{4}-\frac{(x+3)^{2}}{9}=1 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ \frac{(x-3)^{2}}{64}+\frac{(y+4)^{2}}{81}=1 $$

In Problems \(11-16,\) use rotation of axes to eliminate the \(x y\) -term in the given equation. Identify the conic and graph. $$ 2 x^{2}-3 x y-2 y^{2}=5 $$

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \((x-2)^{2}+y=0\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ 25(x-3)^{2}-5(y-1)^{2}=125 $$

Find the center, foci, vertices, endpoints of the minor axis, and eccentricity of the given ellipse. Graph the ellipse. $$ 4 x^{2}+\left(y+\frac{1}{2}\right)^{2}=4 $$

In Problems \(11-16,\) use rotation of axes to eliminate the \(x y\) -term in the given equation. Identify the conic and graph. $$ x^{2}-2 x y+y^{2}=8 x+8 y $$

Consider the point \(P(-2,5,4)\) (a) If lines are drawn from \(P\) perpendicular to the coordinate planes, what are the coordinates of the point at the base of each perpendicular? (b) If a line is drawn from \(P\) to the plane \(z=-2\), what are the coordinates of the point at the base of the perpendicular? (c) Find the point in the plane \(x=3\) that is closest to \(P\).

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(y^{2}+12 y-4 x+16=0\)

In Problems \(1-20\), find the center, foci, vertices, asymptotes, and eccentricity of the given hyperbola. Graph the hyperbola. $$ 10(x+1)^{2}-2\left(y-\frac{1}{2}\right)^{2}=100 $$