Chapter 11

Exer. 33-36: Find an equation for the set of points in an \(x y\)-plane such that the sum of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$ F(12,0), \quad F^{\prime}(-12,0) ; \quad k=26 $$

Exer. 35-36: An ellipse has a focus at the pole with the given center \(C\) and vertex \(V\). Find (a) the eccentricity and (b) a polar equation for the ellipse. $$ C\left(3, \frac{\pi}{2}\right), V\left(1, \frac{3 \pi}{2}\right) $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+6 x-y^{2}=7$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r^{2}\left(\cos ^{2} \theta+4 \sin ^{2} \theta\right)=16 $$

Exer. 35-38: Find an equation for the indicated half of the parabola. $$ \text { Lower half of }(y+1)^{2}=x+3 $$

Exer. 33-36: Find an equation for the set of points in an \(x y\)-plane such that the sum of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$ F(0,15), \quad F^{\prime}(0,-15) ; \quad k=34 $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$5 x^{2}+4 x+4 y^{2}-24 y=-36$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r^{2} \sin 2 \theta=4 $$

Exer. 35-38: Find an equation for the indicated half of the parabola. $$ \text { Upper half of }(y-2)^{2}=x-4 $$

Exer. 33-36: Find an equation for the set of points in an \(x y\)-plane such that the sum of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$ F(0,8), \quad F^{\prime}(0,-8) ; \quad k=20 $$

Kepler's first law asserts that planets travel in elliptical orbits with the sun at one focus. To find an equation of an orbit, place the pole \(O\) at the center of the sun and the polar axis along the major axis of the ellipse (see the figure). (a) Show that an equation of the orbit is $$ r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta} $$ where \(e\) is the eccentricity and \(2 a\) is the length of the major axis. (b) The perihelion distance \(r_{\text {per }}\) and aphelion distance \(r_{\text {aph }}\) are defined as the minimum and maximum distances, respectively, of a planet from the sun. Show that $$ r_{\text {per }}=a(1-e) \quad \text { and } \quad r_{\mathrm{aph}}=a(1+e) $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$-x^{2}=y^{2}-25$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r(\sin \theta-2 \cos \theta)=6 $$

Exer. 35-38: Find an equation for the indicated half of the parabola. $$ \text { Right half of }(x+1)^{2}=y-4 $$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r(3 \cos \theta-4 \sin \theta)=12 $$

Exer. 35-38: Find an equation for the indicated half of the parabola. $$ \text { Left half of }(x+3)^{2}=y+2 $$

A circle \(C\) of radius \(b\) rolls on the outside of the circle \(x^{2}+y^{2}=a^{2}\), and \(b

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$4 x^{2}-16 x+9 y^{2}+36 y=-16$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r\left(\sin \theta+r \cos ^{2} \theta\right)=1 $$

Exer. 39-40: Find an equation for the parabola that has a vertical axis and passes through the given points. $$ P(2,5), \quad Q(-2,-3), \quad R(1,6) $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ y=11 \sqrt{1-\frac{x^{2}}{49}} $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x+4=y^{2}+y$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r\left(r \sin ^{2} \theta-\cos \theta\right)=3 $$

Exer. 39-40: Find an equation for the parabola that has a vertical axis and passes through the given points. $$ P(3,-1), \quad Q(1,-7), \quad R(-2,14) $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ y=-6 \sqrt{1-\frac{x^{2}}{25}} $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$x^{2}+3 x=3 y-6$$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=8 \sin \theta-2 \cos \theta $$

Exer. 41-42: Find an equation for the parabola that has a horizontal axis and passes through the given points. $$ P(-1,1), \quad Q(11,-2), \quad R(5,-1) $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ x=-\frac{1}{3} \sqrt{9-y^{2}} $$

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$9 x^{2}-y^{2}=10-2 y$$

Exer. 41-42: Find an equation for the parabola that has a horizontal axis and passes through the given points. $$ P(2,1), \quad Q(6,2), \quad R(12,-1) $$

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=2 \cos \theta-4 \sin \theta $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ x=\frac{4}{5} \sqrt{25-y^{2}} $$

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{c} y^{2}-4 x^{2}=16 \\ y-x=4 \end{array}\right. $$

A mirror for a reflecting telescope has the shape of a (finite) paraboloid of diameter 8 inches and depth 1 inch. How far from the center of the mirror will the incoming light collect? Exercise 43

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=\tan \theta $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ x=1+2 \sqrt{1-\frac{(y+2)^{2}}{9}} $$

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection. $$ \left\\{\begin{array}{l} x^{2}-y^{2}=4 \\ y^{2}-3 x=0 \end{array}\right. $$

A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the receiver be placed to receive the greatest intensity of sound waves?

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=6 \cot \theta $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ x=-2-5 \sqrt{1-\frac{(y-1)^{2}}{16}} $$

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(13,0), \quad F^{\prime}(-13,0) ; \quad k=24$$

A searchlight reflector has the shape of a paraboloid, with the light source at the focus. If the reflector is 3 feet across at the opening and 1 foot deep, where is the focus?

Exer. 45-78: Sketch the graph of the polar equation. $$ r=5 $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ y=2-7 \sqrt{1-\frac{(x+1)^{2}}{9}} $$

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(5,0), \quad F^{\prime}(-5,0) ; \quad k=8$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-2 $$

Exer. 39-46: Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse. $$ y=-1+\sqrt{1-\frac{(x-3)^{2}}{16}} $$

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(0,10), \quad F^{\prime}(0,-10) ; \quad k=16$$

A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the width of the dish if the depth is to be 2 feet.