Chapter 10

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(y^{2}-x^{2}+2 x-6 y-8=0\)

Convert the polar equation to rectangular form. $$r=4 \sin \theta$$

Determine whether the statement is true or false. Justify your answer. The two sets of parametric equations \(x=t, y=t^{2}+1\) and \(x=3 t, \quad y=9 t^{2}+1\) correspond to the same rectangular equation.

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$x^{2}+6 y=0$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}-6 x-2 y+7=0\)

Determine whether the equation represents a degenerate conic. Explain. $$16 x^{2}+25 y^{2}-32 x+50 y+16=0$$

Convert the polar equation to rectangular form. $$r=2 \cos \theta$$

Determine whether the statement is true or false. Justify your answer. Because the graphs of the parametric equations \(x=t^{2}\) \(y=t^{2}\) and \(x=t, y=t\) both represent the line \(y=x\) they are the same plane curve.

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$x+y^{2}=0$$

Determine whether the equation represents a degenerate conic. Explain. $$9 x^{2}+25 y^{2}-36 x-50 y+61=0$$

The graph of \(r=f(\theta)\) is rotated about the pole through an angle \(\phi .\) Show that the equation of the rotated graph is \(r=f(\theta-\phi)\).

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$(x+1)^{2}-8(y+2)=0$$

Determine whether the statement is true or false. Justify your answer. In the standard form of the equation of a hyperbola, the larger the ratio of \(b\) to \(a\), the larger the eccentricity of the hyperbola.

It Is the ellipse \(\frac{x^{2}}{328}+\frac{y^{2}}{327}=1\) better described as elongated or nearly circular? Explain your reasoning.

Determine whether the statement is true or false. Justify your answer. The parametric equations \(x=a t+h\) and \(y=b t+k\) where \(a \neq 0\) and \(b \neq 0,\) represent a circle centered at \((h, k)\) when \(a=b.\)

Consider the graph of \(r=f(\sin \theta)\). (a) Show that when the graph is rotated counterclockwise \(\pi / 2\) radians about the pole, the equation of the rotated graph is \(r=f(-\cos \theta)\). (b) Show that when the graph is rotated counterclockwise \(\pi\) radians about the pole, the equation of the rotated graph is \(r=f(-\sin \theta)\). (c) Show that when the graph is rotated counterclockwise \(3 \pi / 2\) radians about the pole, the equation of the rotated graph is \(r=f(\cos \theta)\).

Convert the polar equation to rectangular form. $$\theta=\pi / 2$$

Show that the polar equation of the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta}$$

The curve shown is represented by the parametric equations $$x=6 \cos \theta \text { and } y=6 \sin \theta, \quad 0 \leq \theta \leq 6.$$ (a) Describe the orientation of the curve. (b) Determine a range of \(\theta\) that gives the graph of a circle. (c) Write a set of parametric equations representing the curve so that the curve traces from the same point as the original curve but in the opposite direction. (d) How does the original curve change when cosine and sine are interchanged?

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$\left(x+\frac{3}{2}\right)^{2}=4(y-2)$$

Determine whether the statement is true or false. Justify your answer. If \(D \neq 0\) and \(E \neq 0,\) then the graph of \(x^{2}-y^{2}+D x+E y=0\) is a hyperbola.

Show that the polar equation of the hyperbola $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{-b^{2}}{1-e^{2} \cos ^{2} \theta}$$.

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$\left(x+\frac{1}{2}\right)^{2}=4(y-1)$$

It Find the equation of an ellipse such that for any point on the ellipse, the sum of the distances from the points (2,2) and (10,2) is 36.

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$f(x)=\frac{4 x^{2}}{x^{2}+1}$$

Find the zeros (if any) of the rational function. $$f(x)=\frac{x^{2}-9}{x+1}$$

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y^{2}+6 y+8 x+25=0$$

Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form.

Show that \(a^{2}=b^{2}+c^{2}\) for the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ where \(a>0, b>0,\) and the distance from the center of the ellipse (0,0) to a focus is \(c .\)

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$f(x)=\sqrt{x}$$

Find the zeros (if any) of the rational function. $$f(x)=6+\frac{4}{x^{2}+4}$$

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y^{2}-4 y-4 x=0$$

Determine whether the sequence is arithmetic, geometric, or neither. $$66,55,44,33,22, \dots$$

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$y=e^{x}$$

Find the zeros (if any) of the rational function. $$f(x)=5-\frac{3}{x-2}$$

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$x^{2}+4 x+6 y-2=0$$

Find the equation of the hyperbola for any point at which the difference between its distances from the points (2,2) and (10,2) is 6

Determine whether the sequence is arithmetic, geometric, or neither. $$80,40,20,10,5, \ldots$$

Check for symmetry with respect to both axes and to the origin. Then determine whether the function is even, odd, or neither. $$(x-2)^{2}=y+4$$

Find the zeros (if any) of the rational function. $$f(x)=\frac{x^{3}-27}{x^{2}+4}$$

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$x^{2}-2 x+8 y+9=0$$

Determine whether the sequence is arithmetic, geometric, or neither. $$\frac{1}{4}, \frac{1}{2}, 1,2,4, \dots$$

Convert the polar equation to rectangular form. $$r=2 \sin 3 \theta$$

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y^{2}+x+y=0$$

Prove that the graph of the equation \(A x^{2}+C y^{2}+D x+E y+F=0\) is one of the following (except in degenerate cases). (a) Circle (b) Parabola (c) Ellipse \(A=c\) \(A=0\) or \(C=0\) (but not both) \(A C>0\) \(\Delta C<0\)

Determine whether the sequence is arithmetic, geometric, or neither. $$-\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \frac{7}{2}, \dots$$

Convert the polar equation to rectangular form. $$r=-3 \cos 2 \theta$$

Given the hyperbolas \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1 \quad\) and \(\quad \frac{y^{2}}{9}-\frac{x^{2}}{16}=1\) describe any common characteristics that the hyperbolas share, as well as any differences in the graphs of the hyperbolas. Verify your results by using a graphing utility to graph both hyperbolas in the same viewing window.

Find the sum. $$\sum_{n=0}^{6} 3^{n}$$

Convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$