Chapter 10

Find the focus and directrix of the parabola $$ 2 y^{2}-4 y-10 x=0 $$

Find the total area of the rose \(r=a \cos n \theta\), where \(n\) is a positive integer.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=e^{\theta}, \theta \geq 0\) (logarithmic spiral)

Find the equation of the tangent line to the parabola \(y^{2}=-18 x\) that is parallel to the line \(3 x-2 y+4=0\).

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{1+2 \sin \theta}\)

In Problems \(21-30\), find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=\frac{2}{1+t^{2}}, y=\frac{2}{t\left(1+t^{2}\right)} ; t \neq 0 $$

Any line segment through the focus of a parabola, with end points on the parabola, is a focal chord. Prove that the tangent lines to a parabola at the end points of any focal chord intersect on the directrix.

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{6}{2+\sin \theta}\)

In Problems 31-34, find the equation of the tangent line to the given curve at the given value of \(t\) without eliminating the parameter. Make a sketch. $$ x=t^{2}, y=t^{3}, t=2 $$

The sum of the distances of \(P\) from \((0, \pm 9)\) is 26 .

Find the foci of the ellipse $$ 16(x-1)^{2}+25(y+2)^{2}=400 $$

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{6}{4-\cos \theta}\)

In Problems 31-34, find the equation of the tangent line to the given curve at the given value of \(t\) without eliminating the parameter. Make a sketch. $$ x=3 t, y=8 t^{3} ; t=-\frac{1}{2} $$

The sum of the distances of \(P\) from \((\pm 4,0)\) is 14 .

Find the focus and directrix of the parabola $$ x^{2}-6 x+4 y+3=0 $$

Assume that a planet of mass \(m\) is revolving around the sun (located at the pole) with constant angular momentum \(m r^{2} d \theta / d t\). Deduce Kepler's Second Law: The line from the sun to the planet sweeps out equal areas in equal times.

In Problems \(1-32\), sketch the graph of the given polar equation and verify its symmetry (see Examples 1-3). \(r=-\frac{1}{\theta}, \theta>0\) (reciprocal spiral)

A chord of a parabola that is perpendicular to the axis and 1 unit from the vertex has length 1 unit. How far is it from the vertex to the focus?

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{2+2 \cos \theta}\)

Find the equation of the given conic. Horizontal ellipse with center \((5,1)\), major diameter 10 , minor diameter 8

In Problems 33-38, sketch the given curves and find their points of intersection. \(r=6, r=4+4 \cos \theta\)

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{2+2 \cos (\theta-\pi / 3)}\)

In Problems 31-34, find the equation of the tangent line to the given curve at the given value of \(t\) without eliminating the parameter. Make a sketch. $$ x=2 e^{t}, y=\frac{1}{3} e^{-t} ; t=0 $$

The difference of the distances of \(P\) from \((0, \pm 6)\) is 10 .

Find the equation of the given conic. Hyperbola with center \((2,-1)\), vertex at \((4,-1)\), and focus at \((5,-1)\)

In Problems 33-38, sketch the given curves and find their points of intersection. \(r=1-\cos \theta, r=1+\cos \theta\)

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{\frac{1}{2}+\cos (\theta-\pi)}\)

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=2 t-1, y=3 t-4 ; 0 \leq t \leq 3 $$

Find the equation of the tangent line to the given curve at the given point. \(\frac{x^{2}}{27}+\frac{y^{2}}{9}=1\) at \((3, \sqrt{6})\)

Find the equation of the given conic. Parabola with vertex \((2,3)\) and focus \((2,5)\)

In Problems 33-38, sketch the given curves and find their points of intersection. \(r=3 \sqrt{3} \cos \theta, r=3 \sin \theta\)

In Problems 23-36, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. \(r=\frac{4}{3 \cos (\theta-\pi / 3)}\)

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=2-t, y=2 t-3 ;-3 \leq t \leq 3 $$

Find the equation of the tangent line to the given curve at the given point. \(\frac{x^{2}}{24}+\frac{y^{2}}{16}=1\) at \((3 \sqrt{2},-2)\)

In Problems 33-38, sketch the given curves and find their points of intersection. \(r=5, r=\frac{5}{1-2 \cos \theta}\)

Show that the polar equation of the circle with center \((c, \alpha)\) and radius \(a\) is \(r^{2}+c^{2}-2 r c \cos (\theta-\alpha)=a^{2}\).

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=t, y=t^{3 / 2} ; 0 \leq t \leq 3 $$

Find the equation of the tangent line to the given curve at the given point. \(\frac{x^{2}}{27}+\frac{y^{2}}{9}=1\) at \((3,-\sqrt{6})\)

Find the equation of the given conic. Hyperbola with vertices at \((0,0)\) and \((0,6)\) and a focus at \((0,8)\)

In Problems 33-38, sketch the given curves and find their points of intersection. \(r=6 \sin \theta, r=\frac{6}{1+2 \sin \theta}\)

Prove that \(r=a \sin \theta+b \cos \theta\) represents a circle and find its center and radius.

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=2 \sin t, y=2 \cos t ; 0 \leq t \leq \pi $$

Find the equation of the tangent line to the given curve at the given point. \(\frac{x^{2}}{2}-\frac{y^{2}}{4}=1\) at \((\sqrt{3}, \sqrt{2})\)

Find the equation of the given conic. Ellipse with foci at \((2,0)\) and \((2,12)\) and a vertex at \((2,14)\)

In Problems 33-38, sketch the given curves and find their points of intersection. \(r^{2}=4 \cos 2 \theta, r=2 \sqrt{2} \sin \theta\)

Find the length of the latus rectum for the general conic \(r=e d /\left[1+e \cos \left(\theta-\theta_{0}\right)\right]\) in terms of \(e\) and \(d\).

In Problems 35-46, find the length of the parametric curve defined over the given interval. $$ x=3 t^{2}, y=t^{3} ; 0 \leq t \leq 2 $$

Find the equation of the tangent line to the given curve at the given point. \(x^{2}+y^{2}=169\) at \((5,12)\)

Find the equation of the given conic. Parabola with focus \((2,5)\) and directrix \(x=10\)

Plot the curve \(r=4 \sin (3 \theta / 2), 0 \leq \theta \leq 4 \pi\), and then find its length.