Chapter 22

Even though you may be able to solve some of these with only the Pythagorean theorem, we suggest that you use analytic geometry for the practice. Prove that any angle inscribed in a semicircle is a right angle.

Rewrite standard equation in general form. $$(y-5)^{2}=9(x+8)$$

Slope Find the slope of each straight line. Connecting (2,4) and (5,7)

Write the equation of each ellipse.Center at (2,-1)\(;\) a vertex at (2,5)\(;\) length of minor axis \(=3\).

Write each standard equation in general form. $$\frac{(x+5)^{2}}{9}-\frac{(y+7)^{2}}{4}=22$$

Write the equation of each line in general form. passes through (4,-1)\(;\) parallel to \(4 x-y=-3\)

Slope Find the slope of each straight line. Connecting (5,2) and (3,6)

Write each general equation in standard form by completing the square. Find the center, lengths of the semiaxes, foci, slope of the asymptotes, and vertices. $$16 x^{2}-64 x-9 y^{2}-54 y=161$$

Write the equation of each line in general form. passes through (-4,2)\(;\) perpendicular to \(y=5 x-3\)

Slope Find the slope of each straight line. Connecting (-2,5) and (5,-6)

Write each standard equation in general form. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$

Write the equation of each line in general form. passes through (6,1)\(;\) perpendicular to \(6 y-2 x=3\)

Slope Find the slope of each straight line. Connecting (3,-3) and (-6,2)

Write the equation of each line in general form. passes through (5,2)\(;\) is parallel to the \(x\) axis.

Angle of Inclination Find the slope of the line having the given angle of inclination. $$38.2^{\circ}$$

Write the equation of each line in general form. passes through (-3,6)\(;\) is parallel to the \(y\) axis.

Angle of Inclination Find the slope of the line having the given angle of inclination. $$77.9^{\circ}$$

Write the equation of each line in general form. passes through points (3,5) and (-1,2)

Angle of Inclination Find the slope of the line having the given angle of inclination. $$1.83 \mathrm{rad}$$

Write the equation of a hyperbola with center at the origin and with asymptotes on the coordinate axes, which passes through the point (-9,2)

Write each general equation in standard form. Find the center, foci, vertices, and the lengths \(a\) and \(b\) of the semi axes, and make a graph.$$4 x^{2}+y^{2}-16 x+6 y+21=0$$

Write the equation of each line in general form. passes through points (4.24,-1.25) and (3.85,4.27)

Angle of Inclination Find the slope of the line having the given angle of inclination. $$58^{\circ} 14^{\prime}$$

Write each general equation in standard form. Find the center, foci, vertices, and the lengths \(a\) and \(b\) of the semi axes, and make a graph.$$x^{2}+9 y^{2}-4 x+18 y+4=0$$

Write the equation of each line in general form. \(x\) intercept \(=5 ; y\) intercept \(=-3\)

Angle of Inclination Find the slope of the line having the given angle of inclination. $$156.3^{\circ}$$

Write each general equation in standard form. Find the center, foci, vertices, and the lengths \(a\) and \(b\) of the semi axes, and make a graph.$$16 x^{2}+25 y^{2}+96 x-50 y-231=0$$

Write the equation of each line in general form. \(x\) intercept \(=-2 ; y\) intercept \(=6\)

Angle of Inclination Find the slope of the line having the given angle of inclination. $$132.8^{\circ}$$

Write each general equation in standard form. Find the center, foci, vertices, and the lengths \(a\) and \(b\) of the semi axes, and make a graph.$$49 x^{2}+81 y^{2}+294 x+810 y-1503=0$$

Rewrite each equation in general form. $$y=3 x+2$$

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=3$$

Rewrite each equation in general form. $$y=4 x-5$$

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=1.84$$

Rewrite each equation in general form. $$y=-2 x+6$$

A parabolic curve is to be used at a dip in a highway. The road dips \(32.0 \mathrm{m}\) in a horizontal distance of \(125 \mathrm{m}\) and then rises to its previous height in another \(125 \mathrm{m}\) Write the equation of the curve of the roadway, taking the origin at the bottom of the dip and the \(y\) axis vertical.

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=-4$$

First write the equation for each ellipse.A certain bridge arch is in the shape of half an ellipse \(120 \mathrm{ft}\) wide and \(30.0 \mathrm{ft}\) high. At what horizontal distance from the center of the arch is the height equal to \(15.0 \mathrm{ft} ?\)

Rewrite each equation in general form. $$y=-4 x-3$$

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=-2.75$$

Rewrite each equation in slope-intercept form. $$x+2 y+5=0$$

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=0$$

The paths of the planets and certain comets are ellipses, with the sun at one focal point. The path of Halley's comet is an ellipse with a major axis of 36.18 AU and a minor axis of 9.12 AU. An astronomical unit, \(\mathrm{AU}\), is the distance between the earth and the sun, about 92.6 million miles. What is the greatest distance that Halley's comet gets from the sun?

Rewrite each equation in slope-intercept form. $$3 x+4 y-6=0$$

Find the angle of inclination, in decimal degrees to three significant digits, of a line having the given slope. $$m=-15$$

Rewrite each equation in slope-intercept form. $$-7 x+3 y+8=0$$

Find the angle of inclination, in decimal degrees to three significant digits, of a line passing through the given points. (5,2) and (-3,4)

Rewrite each equation in slope-intercept form. $$-2 x+2 y-9=0$$

Project: If a projectile is launched with a horizontal velocity \(v_{x}\) and vertical velocity \(v_{y},\) its position after \(t\) seconds is given by $$\begin{array}{l}x=v_{x} t \\\y=v_{y} t-(g / 2) t^{2}\end{array}$$ where \(g\) is the acceleration due to gravity. Eliminate \(t\) from this pair of equations to get \(y=f(x),\) and show that this equation represents a parabola.

Find the angle of inclination, in decimal degrees to three significant digits, of a line passing through the given points. (-2.5,-3.1) and (5.8,4.2)