Chapter 11

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 10),\) vertices \((0, \pm 8)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focus \(F(5,0)\)

Show by eliminating the parameter \(\theta\) that the following parametric equations represent a hyperbola: $$ x=a \tan \theta \quad y=b \sec \theta $$

Find an equation for the ellipse that satisfies the given conditions. Foci \(( \pm 4,0),\) vertices \(( \pm 5,0)\)

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$6 x^{2}+10 x y+3 y^{2}-6 y=36$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ 3 x^{2}+4 y^{2}-6 x-24 y+39=0 $$

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 2),\) vertices \((0, \pm 1)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(x=2\)

Find an equation for the ellipse that satisfies the given conditions. Foci \((0, \pm 3),\) vertices \((0, \pm 5)\)

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$9 x^{2}-6 x y+y^{2}+6 x-2 y=0$$

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$ x^{2}+4 y^{2}+20 x-40 y+300=0 $$

Find an equation for the hyperbola that satisfies the given conditions. Foci \(( \pm 6,0),\) vertices \(( \pm 2,0)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(y=6\)

Sketch the curve given by the parametric equations. $$ x=t \cos t, \quad y=t \sin t, \quad t \geq 0 $$

Find an equation for the ellipse that satisfies the given conditions. Length of major axis \(4,\) length of minor axis \(2,\) foci on \(y\) -axis

(a) Use rotation of axes to show that the following equation represents a hyperbola: $$7 x^{2}+48 x y-7 y^{2}-200 x-150 y+600=0$$ (b) Find the \(X Y-\) and \(x y\) -coordinates of the center, vertices, and foci. (c) Find the equations of the asymptotes in \(X Y\) - and \(x y\) -coordinates.

Use a graphing device to graph the conic. $$ 2 x^{2}-4 x+y+5=0 $$

Find an equation for the hyperbola that satisfies the given conditions. Vertices \(( \pm 1,0),\) asymptotes \(y=\pm 5 x\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(y=-10\)

Sketch the curve given by the parametric equations. $$ x=\sin t, \quad y=\sin 2 t $$

Find an equation for the ellipse that satisfies the given conditions. Length of major axis \(6,\) length of minor axis \(4,\) foci on \(x\) -axis

(a) Use rotation of axes to show that the following equation represents a parabola: $$2 \sqrt{2}(x+y)^{2}=7 x+9 y$$ (b) Find the \(X Y\) - and \(x y\) -coordinates of the vertex and focus. (c) Find the equation of the directrix in \(X Y\) - and \(x y\) -coordinates.

Use a graphing device to graph the conic. $$ 4 x^{2}+9 y^{2}-36 y=0 $$

Find an equation for the hyperbola that satisfies the given conditions. Vertices \((0, \pm 6),\) asymptotes \(y=\pm \frac{1}{3} x\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(x=-\frac{1}{8}\)

Sketch the curve given by the parametric equations. $$ x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}} $$

Find an equation for the ellipse that satisfies the given conditions. Foci \((0, \pm 2),\) length of minor axis 6

Solve the equations: $$x=X \cos \phi-Y \sin \phi$$ $$y=X \sin \phi+Y \cos \phi$$ for \(X\) and \(Y\) in terms of \(x\) and \(y .\) \([\) Hint: To begin, multiply the first equation by cos \(\phi\) and the second by \(\sin \phi,\) and then add the two equations to solve for \(X . ]\)

Use a graphing device to graph the conic. $$ 9 x^{2}+36=y^{2}+36 x+6 y $$

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 8),\) asymptotes \(y=\pm \frac{1}{2} x\)

Sketch the curve given by the parametric equations. $$ x=\cot t, \quad y=2 \sin ^{2} t, \quad 0

Find an equation for the ellipse that satisfies the given conditions. Foci \(( \pm 5,0),\) length of major axis 12

Show that the graph of the equation $$\sqrt{x}+\sqrt{y}=1$$ is part of a parabola by rotating the axes through an angle of \(45^{\circ}\).[Hint: First convert the equation to one that does not involve radicals.]

Use a graphing device to graph the conic. $$ x^{2}-4 y^{2}+4 x+8 y=0 $$

Find an equation for the hyperbola that satisfies the given conditions. Vertices \((0, \pm 6),\) hyperbola passes through \((-5,9)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix has \(y\) -intercept 6

If a projectile is fired with an initial speed of \(v_{0} \mathrm{ft} / \mathrm{s}\) at an angle \(\alpha\) above the horizontal, then its position after \(t\) seconds is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-16 t^{2} $$ (where \(x\) and \(y\) are measured in feet). Show that the path of the projectile is a parabola by eliminating the parameter \(t\)

Find an equation for the ellipse that satisfies the given conditions. Endpoints of major axis \(( \pm 10,0),\) distance between foci 6

Let \(Z, Z^{\prime},\) and \(R\) be the matrices $$Z=\left[\begin{array}{l}{x} \\ {y}\end{array}\right] \quad Z^{\prime}=\left[\begin{array}{l}{X} \\ {Y}\end{array}\right]$$ $$R=\left[\begin{array}{cc}{\cos \phi} & {-\sin \phi} \\ {\sin \phi} & {\cos \phi}\end{array}\right]$$ Show that the Rotation of Axes Formulas can be written as $$Z=R Z^{\prime} \quad \text { and } \quad Z^{\prime}=R^{-1} Z$$

Determine what the value of \(F\) must be if the graph of the equation $$ 4 x^{2}+y^{2}+4(x-2 y)+F=0 $$ is (a) an ellipse, (b) a single point, or (c) the empty set.

Find an equation for the hyperbola that satisfies the given conditions. Asymptotes \(y=\pm x,\) hyperbola passes through \((5,3)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Opens upward with focus 5 units from the vertex

Find an equation for the ellipse that satisfies the given conditions. Endpoints of minor axis \((0, \pm 3),\) distance between foci 8

Find an equation for the ellipse that shares a vertex and a focus with the parabola \(x^{2}+y=100\) and has its other focus at the origin.

Find an equation for the hyperbola that satisfies the given conditions. Foci \(( \pm 3,0),\) hyperbola passes through \((4,1)\)

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Focal diameter 8 and focus on the negative \(y\) -axis

Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin t, \quad y=2 \cos 3 t $$

Find an equation for the ellipse that satisfies the given conditions. Length of major axis \(10,\) foci on \(x\) -axis, ellipse passes through the point \((\sqrt{5}, 2)\)

Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance \(d(P, Q)\) and \(d\left(P^{\prime}, Q^{\prime}\right)\) where \(P^{\prime}\) and \(Q^{\prime}\) are the images of \(P\) and \(Q\) under a rotation of axes.

This exercise deals with confocal parabolas, that is, families of parabolas that have the same focus. (a) Draw graphs of the family of parabolas $$x^{2}=4 p(y+p)$$ for \(p=-2,-\frac{3}{2},-1,-\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2\) (b) Show that each parabola in this family has its focus at the origin. (c) Describe the effect on the graph of moving the vertex closer to the origin.