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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Find the equation of each of the lines with the given properties. Sketch the graph of each line.Passes through (5.3,−2.7) and is parallel to the x-axis.
Find the distance between the given pairs of points.(√32, √18) and (−√50, √8)
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.x2 + 2xy + y2 − 2x + 2y = 0
Plot the given polar coordinate points on polar coordinate paper.(−3, −5π/4)
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Straight line: passes through (1,−7) with a slope of 4
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.x2 = −4y
Plot the curves of the given polar equations in polar coordinates.r = 2sinθ
Find the equation of each of the curves described by the given information.Parabola: vertex (−1, 3), focus (−1, 6)
Find the equation of each of the circles from the given information.Center at ,(3/2 − 2), radius 5/2
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.y2 = 25(1 − x2)
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.3.2x2 = 2.1y(1 − 2y)
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.9y2 − 16x2 = 9
(a) What type of curve is represented by 8x2 − 4xy + 5y2 = 36?(b) Through what angle must the curve in part (a) be rotated in order that there is no x'y' -term?
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Passes through (−15, 9) and is perpendicular to the x-axis.
Find the distance between the given pairs of points.(e,− π) and (−2e, −π)
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.5x2 − 6xy + 5y2 = 32
Plot the given polar coordinate points on polar coordinate paper.(−4, −5π/3)
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Straight line: passes through (−1, 5) and (−2,−3)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.x2 + 12y = 0
Plot the curves of the given polar equations in polar coordinates.r = −3cosθ
Find the equation of each of the curves described by the given information.Parabola: focus (4,−4), directrix y = −2
Find the equation of each of the circles from the given information.Center at (12,−15), radius 18
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.y2 = 8(2 − x2)
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.36x2 = 12y(1 − 3y) + 1
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.y2 = 4(x2 + 1)
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Is parallel to the y-axis and is 3 units to the left of it.
Find the distance between the given pairs of points.(1.22,−3.45) and (−1.07,−5.16)
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.xy = 8
Plot the given polar coordinate points on polar coordinate paper.(2, 2)
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Straight line: perpendicular to 3x − 2y + 8 = 0 and has a y-intercept of (0,−1)
Plot the curves of the given polar equations in polar coordinates.1 − r = cosθ (cardioid)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.2y2 − 5x = 0
Find the equation of each of the curves described by the given information.Parabola: axis, directrix are coordinate axes, focus (10, 0)
Find the equation of each of the circles from the given information.Center at (−3,−5), radius 2√3
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.2x2 + 3y2 = 600
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.y = 3(1 − 2x)(1 + 2x)
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.y2 = 9(x2 − 1)
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Is parallel to the x-axis and is 4.1 units below it.
Find the distance between the given pairs of points.(a, h2) and [a + h, (a + h)2]
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.5x2 − 8xy + 5y2 = 0
Plot the given polar coordinate points on polar coordinate paper.(−6,−6)
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Straight line: parallel to 2x − 5y + 1 = 0 and has an x-intercept of (2, 0)
Plot the curves of the given polar equations in polar coordinates.r + 1 = sinθ (cardioid)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.3x2 + 8y = 0
Find the equation of each of the curves described by the given information.Parabola: vertex (4, 4), vertical directrix, passes through (0, 1)
Find the equation of each of the curves described by the given information.Ellipse: center (−2, 2), focus (−5, 2), vertex (−7, 2)
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.4x2 + 25y2 = 0.25
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.y(3 − 2x) = x(5 − 2y)
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.4x2 − y2 = 0.64
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.3x2 + 4xy = 4
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Perpendicular to line with slope of −3; passes through (1,−2).
Find the slopes of the lines through the points.(3, 8) and (−1,−2)
Plot the given polar coordinate points on polar coordinate paper.(0.5,−8.4)
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Circle: concentric with x2 + y2 = 6x, passes through (4,−3)
Plot the curves of the given polar equations in polar coordinates.r = 2 − cosθ (limaçon)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.y = 0.48x2
Find the equation of each of the curves described by the given information.Ellipse: center (0, 3), focus (12, 3), major axis 26 units
Find the equation of each of the circles from the given information.The points (3, 8) and (−3, 0) are the ends of a diameter.
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve.9x2 + 4y2 = 0.09
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.x(13 − 5x) = 5y2
Find the coordinates of the vertices and the foci of the given hyperbolas. Sketch each curve.9y2 − x2 = 0.72
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Parallel to line through (−1, 7) and (3, 1); passes through (−3,−4).
Find the slopes of the lines through the points.(−1, 3) and (−8,−4)
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.9x2 − 24xy + 16y2 − 320x − 240y = 0
Plot the given polar coordinate points on polar coordinate paper.(−2.2, 18.8)
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Circle: tangent to lines x = 3 and x = 9, center on line y = 2x
Plot the curves of the given polar equations in polar coordinates.r = 2 + 3sinθ (limaçon)
Determine the coordinates of the focus and the equation of the directrix of the given parabolas. Sketch each curve.x = −7.6y2
Find the equation of each of the curves described by the given information.Ellipse: center (−2, 1), vertex (−2, 5), passes through (0, 1)
Find the equation of each of the circles from the given information.Concentric with the circle (x − 2)2 + (y − 1)2 = 4 and passes through (4,−1)
Find the equations of the ellipses satisfying the given conditions. The center of each is at the origin.Vertex (15, 0), focus (9, 0)
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.2xy + x − 3y = 6
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Has equal intercepts and passes through (5, 2).
Find the equations of the hyperbolas satisfying the given conditions. The center of each is at the origin.Vertex (3, 0), focus (5, 0)
Find the slopes of the lines through the points.(4,−5) and (4,−8)
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.11x2 − 6xy + 19y2 = 20
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Parabola: focus (−3, 0), vertex (0, 0)
Find a set of polar coordinates for each of the points for which the rectangular coordinates are given.(√3, 1)
Plot the curves of the given polar equations in polar coordinates.r = 4sin 2θ (rose)
Find the equations of the parabolas satisfying the given conditions. The vertex of each is at the origin.Focus (3, 0)
Find the equations of the parabolas satisfying the given conditions. The vertex of each is at the origin.Focus (0, 0.4)
Plot the curves of the given polar equations in polar coordinates.r = 2sin 3θ (rose)
Find the equation of each of the curves described by the given information.Ellipse: foci (1,−2) and (1, 10), minor axis 5 units
Find the equations of the ellipses satisfying the given conditions. The center of each is at the origin.Minor axis 8, vertex (0,−5)
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.(y + 1)2 = x2 + y2 − 1
Find the equation of each of the lines with the given properties. Sketch the graph of each line.Is perpendicular to the line 6.0x − 2.4y − 3.9 = 0 and passes through (7.5,−4.7).
Find the equations of the hyperbolas satisfying the given conditions. The center of each is at the origin.Vertex (0,−1), focus (0,−√3)
Find the slopes of the lines through the points.(−3,−7) and (2, 10)
Transform each equation to a form without an xy-term by a rotation of axes. Identify and sketch each curve. Then display each curve on a calculator.x2 + 4xy − 2y2 = 6
Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.Parabola: vertex (0, 0), passes through (1, 1) and (−2, 4)
Find a set of polar coordinates for each of the points for which the rectangular coordinates are given.(−3, 3)
Plot the curves of the given polar equations in polar coordinates.r2 = 4sin 2θ (lemniscate)
Find the equation of each of the curves described by the given information.Hyperbola: vertex (−1, 1), focus (−1, 4), center (−1, 2)
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.2x2 + y2 + 8x = 8y
In Eq. (21.34), if A = B = C = 0, D ≠ 0, E ≠ 0, and F ≠ 0, describe the locus of the equation.In Eq. 21.34.Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Use matrices A, B, C, and D to find the indicated matrices. If the operations cannot be performed, explain why.A − B A || 6-3 4 -5 B 1) -1 4 -7 2 -6 11 3 12 -9 -6 D= 79 -6 -4 0 8 16000
In Example 6, change the x + 3 to x + 2 and then perform the synthetic division.Data from Example 6Divide x5 + 2x4 − 4x2 + 3x − 4 by x + 3 using synthetic division.Because the powers of x are in
In Example 9, change the 2x − 3 to 2x + 3 and then determine whether 2x + 3 is a factor.Data from Example 9:By using synthetic division, determine whether 2x − 3 is a factor of 2x3 − 3x2 + 8x
Use synthetic division to perform the division (x3 − 5x2 + 4x − 9) ÷ (x − 3).
Solve the given equation without using a calculator.x3 + 5x2 + 2x − 8 = 0

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