Chapter 10

The slope of the tangent line to the parabola \(y^{2}=5 x\) at a certain point on the parabola is \(\sqrt{5} / 4 .\) Find the coordinates of that point. Make a sketch.

Sketch the graph of the given equation. \(x^{2}-4 x+8 y=0\)

Find the length of the logarithmic spiral \(r=e^{\theta / 2}\) from \(\theta=0\) to \(\theta=2 \pi\)

Sketch the graph of the given polar equation and verify its symmetry (see Examples \(1-3)\). \(r=2 \theta, \theta \geq 0\) (spiral of Archimedes)

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=-4 \cos \theta $$

find \(d y / d x\) and \(d^{2} y / d x^{2}\) without eliminating the parameter. $$ x=\cot t-2, y=-2 \csc t+5 ; 0

Find the equation of the given central conic. Hyperbola with foci \((\pm 4,0)\) and directrices \(x=\pm 1\)

The slope of the tangent line to the parabola \(x^{2}=-14 y\) at a certain point on the parabola is \(-2 \sqrt{7} / 7\). Find the coordinates of that point.

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.

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{4}{1+\cos \theta} $$

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

Find the equation of the given central conic. Hyperbola whose asymptotes are \(x \pm 2 y=0\) and that goes through the point \((4,3)\)

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

Sketch the graph of the strophoid \(r=\sec \theta-2 \cos \theta\), and find the area of its loop.

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{4}{1+2 \sin \theta} $$

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.

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

Consider the two circles \(r=2 a \sin \theta\) and \(r=2 b \cos \theta\), with \(a\) and \(b\) positive. (a) Find the area of the region inside both circles. (b) Show that the two circles intersect at right angles.

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{6}{2+\sin \theta} $$

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

Find the equation of the set of points \(P\) satisfying the given conditions. The sum of the distances of \(P\) from \((0, \pm 9)\) is 26 .

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.

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{6}{4-\cos \theta} $$

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

Find the equation of the set of points \(P\) satisfying the given conditions. The sum of the distances of \(P\) from \((\pm 4,0)\) is 14 .

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

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 tangent line to the given curve at the given value of \(t\) without eliminating the parameter. Make a sketch. $$ x=2 \sec t, y=2 \tan t ; t=-\frac{\pi}{6} $$

Find the equation of the set of points \(P\) satisfying the given conditions. The difference of the distances of \(P\) from \((\pm 7,0)\) is 12 .

Prove that the vertex is the point on a parabola closest to the focus.

Hyperbola with center \((2,-1)\), vertex at \((4,-1)\), and focus at \((5,-1)\)

Sketch the given curves and find their points of intersection. $$ r=1-\cos \theta, r=1+\cos \theta $$

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

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

Find the equation of the set of points \(P\) satisfying the given conditions. The difference of the distances of \(P\) from \((0, \pm 6)\) is 10 .

Parabola with vertex \((2,3)\) and focus \((2,5)\)

Sketch the given curves and find their points of intersection. $$ r=3 \sqrt{3} \cos \theta, r=3 \sin \theta $$

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

, 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. $$ \frac{x^{2}}{27}+\frac{y^{2}}{9}=1 \text { at }(3, \sqrt{6}) $$

Sketch the given curves and find their points of intersection. $$ r=5, r=\frac{5}{1-2 \cos \theta} $$

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

, 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. $$ \frac{x^{2}}{24}+\frac{y^{2}}{16}=1 \text { at }(3 \sqrt{2},-2) $$

Sketch the given curves and find their points of intersection. $$ r=6 \sin \theta, r=\frac{6}{1+2 \sin \theta} $$

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