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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Identify the type of curve for each equation, and then view it on a calculator.4(y2 − 4x − 2) = 5(4y − 5)
Determine whether the given lines are parallel, perpendicular, or neither.5x + 2y − 3 = 0 and 10y = 7 − 4x
Use a calculator to view the ellipse 4x2 + 3y2 + 16x − 18y + 31 = 0.
Find any points of intersection of the ellipse 2x2 + y2 = 17 and the hyperbola y2 − x2 = 5.
Find the polar equation of each of the given rectangular equations.x2 + (y − 2)2 = 4
Find the polar equation of each of the given rectangular equations.x2 − y2 = 0.01
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.2x2 + 2y2 − 24x + 16y + 95 = 0
Find the indicated quantities for each of the given equations. Sketch each curve.x2 − 2y2 + 4x + 4y + 6 = 0, center
Determine whether the lines through the two pairs of points are parallel or perpendicular.(−3, 9) and (4, 4); (9,−1) and (4,−8)
View the curves of the given polar equations on a calculator.r = 0.5sinθ
Find the equation of the parabola with focus (1, 1) and directrix y = 5, by use of the definition. Sketch the curve
Determine the center and radius of each circle. Sketch each circle.9x2 + 9y2 + 18y = 7
Identify the type of curve for each equation, and then view it on a calculator.2(2x2 − y) = 8 − y2
Determine whether the given lines are parallel, perpendicular, or neither.48y − 36x = 71 and 52x = 17 − 39y
Use a calculator to view the ellipse 4x2 + 8y2 + 4x − 24y + 1 = 0.
Find any points of intersection of the hyperbolas x2 − 3y2 = 22 and xy = 5.
Find the polar equation of each of the given rectangular equations.x2 + 4y2 = 4
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.16x2 + 25y2 − 32x + 100y − 284 = 0
Determine whether the lines through the two pairs of points are parallel or perpendicular.(−1,−4) and (2, 3); (−5, 2) and (−19, 8)
Find the indicated quantities for each of the given equations. Sketch each curve.x2 − 2xy + y2 + 4x + 4y = 0, vertex
View the curves of the given polar equations on a calculator.r = 2 secθ + 1
Use a calculator to view the parabola y2 + 2x + 8y + 13 = 0.
Determine the center and radius of each circle. Sketch each circle.2x2 + 2y2 = 4x + 8y + 1
Identify the type of curve for each equation, and then view it on a calculator.4(y2 + 6y + 1) = x(x − 4) − 24
The equation of an ellipse with center (h, k) and major axis parallel to the x-axis isSketch the ellipse that has a major axis of 6, a minor axis of 4, and for which (h, k) is (2,−1) (x -
Determine whether the given lines are parallel, perpendicular, or neither.4.5x − 1.8y = 1.7 and 2.4x + 6.0y = 0.3
Find the equation of the hyperbola with foci (1, 2) and (11, 2), and a transverse axis of 8, by use of the definition. Sketch the curve.
Find the polar equation of each of the given rectangular equations.y2 = 4x
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.9x2 − 16y2 − 18x + 96y − 279 = 0
Determine whether the lines through the two pairs of points are parallel or perpendicular.(−a,−2b) and (3a, 6b); (2a,−6b) and (5a, 0)
Find the indicated quantities for each of the given equations. Sketch each curve.9x2 − 9xy + 21y2 − 15 = 0, center
View the curves of the given polar equations on a calculator.r = 2cos(cos2θ)
Use a calculator to view the parabola y2 − 2x − 6y + 19 = 0.
Determine the center and radius of each circle. Sketch each circle.9x2 + 9y2 = 36x − 12
Identify the type of curve for each equation, and then view it on a calculator.8x + 31 − xy = y(y − 2 − x)
Determine whether the given lines are parallel, perpendicular, or neither.3.5y = 4.3 − 1.5x and 3.6x + 8.4y = 1.7
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.5x2 − 3y2 + 95 = 40x
The equation of an ellipse with center (h, k) and major axis parallel to the y-axis isSketch the ellipse that has a major axis of 8, a minor axis of 6, and for which (h, k) is (1, 3). (y – k)2 (x
Find the equation of the hyperbola with vertices (−2, 4) and (−2,−2), and a conjugate axis of 4, by use of the definition. Sketch the curve.
Find the polar equation of each of the given rectangular equations.x2 + y2 = 6y
Determine the value of k.The distance between (−1,3) and (11, k) is 13.
Plot the given curves in polar coordinates.r = 4(1 + sinθ)
View the curves of the given polar equations on a calculator.r = 3cos 4θ
The equation of a parabola with vertex (h, k) and axis parallel to the x-axis is (y − k)2 = 4 p(x − h). Sketch the parabola for which (h, k) is (2,−3) and p = 2.
Determine whether the circles with the given equations are symmetric to either axis or the origin.x2 + y2 = 100
Find k if the lines 4x − ky = 6 and 6x + 3y + 2 = 0 are parallel.
For the equation x2 + ky2 = a2 , what type of curve is represented if (a) k = 1, (b) k < 0, (c) k > 0 (k ≠ 1)?
Determine the center (or vertex if the curve is a parabola) of the given curve. Sketch each curve.4y2 + 29 = 15x + 12y
Determine the value of k.The points in Exercise 39 are the vertices of a right triangle, with the right angle at (3, k).Data from Exercises 39Points (6,−1), (3, k), and (−3,−7) are on the same
Plot the given curves in polar coordinates.r = 3 sinθ − 4 cosθ
View the curves of the given polar equations on a calculator.r = 1 + 3cosθ − 2sinθ
Determine whether the circles with the given equations are symmetric to either axis or the origin.5x2 + 5y2 − 10x + 20y = 3
Find the equation of the circle that has the focus and the vertex of the parabola x2 = 8y as the ends of a diameter.
Find k such that the line through (k, 2) and (3, 1 − k) is perpendicular to the line x − 2y = 5. Explain your method.
In Eq. (21.34), if A = −C ≠ 0, B = D = E = 0 and F = C, describe the locus of the equation if C > 0.Eq. 21.34.Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Show that the ellipse 5x2 + y2 − 3y − 7 = 0 is symmetric to the y-axis.
The equation of a hyperbola with center (h, k) and transverse axis parallel to the y-axis isSketch the hyperbola that has a transverse axis of 2, a conjugate axis of 8, and for which (h, k) is (5, 0)
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r cosθ = 4
Find the equation of the hyperbola with asymptotes x − y = −1 and x + y = −3 and vertex (3,−1).
Show that the given points are vertices of the given geometric figures.(2, 3), (4, 9), and (−2, 7) are vertices of an isosceles triangle.
Plot the given curves in polar coordinates. 3 sin+ 2 cos
View the curves of the given polar equations on a calculator.r = cosθ + sin 2θ
Find the standard equation of the parabola with vertex (0, 0) and that passes through (2, 2) and (8, 4).
A square is inscribed in the circle x2 + y2 = 32 (all four vertices are on the circle). Find the area of the square.
Find the shortest distance from (4, 1) to the line 4x − 3y + 12 = 0.
A flashlight emits a cone of light onto the floor. Determine the type of curve for the perimeter of the lighted area on the floor depending on the position of the flashlight and cone as described.
Show that the parametric equations x = 2sint and y = 3cost define an ellipse.
Two concentric (same center) hyperbolas are called conjugate hyperbolas if the transverse and conjugate axes of one are, respectively, the conjugate and transverse axes of the other. What is the
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r = −cosθ
The circle x2 + y2 + 4x − 5 = 0 passes through the foci and the ends of the minor axis of an ellipse that has its major axis along the x-axis. Find the equation of the ellipse.
Show that the given points are vertices of the given geometric figures.(−1, 3), (3, 5), and (5, 1) are the vertices of a right triangle.
Plot the given curves in polar coordinates. 1 2(sin0 - 1)
View the curves of the given polar equations on a calculator.2r cosθ + r sinθ = 2
For the point (−2, 5), find the point that is symmetric to it with respect to (a) The x-axis, (b) The y-axis, (c) The origin.
For either standard form of the equation of a parabola, describe what happens to the shape of the parabola as |p| increases.
Determine the type of curve for the perimeter of the lighted area on the floor depending on the position of the flashlight and cone as described. See Fig. 21.96.The flashlight is directed toward the
Find the acute angle between the lines x + y = 3 and 2x − 5y = 4.
For the ellipse given byfind the length of the line segment perpendicular to the major axis that passes through a focus and spans the width of the ellipse. X 25 9 1,
As with an ellipse, the eccentricity e of a hyperbola is defined as e = c/a. Find the eccentricity of the hyperbola 2x2 − 3y2 = 24.
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation. 2 cos 3sin0
The vertex and focus of one parabola are, respectively, the focus and vertex of a second parabola. Find the equation of the first parabola, if y2 = 4x is the equation of the second.
Show that the given points are vertices of the given geometric figures.(−5,−4), (7, 1), (10, 5), and (−2, 0) are the vertices of a parallelogram.
Plot the given curves in polar coordinates.r = 2sin(θ/2)
What is the graph of tan θ = 1? Verify by changing the equation to rectangular form.
A television satellite dish measures 80.0 cm across its opening and is 12.5 cm deep. Find the distance between the vertex and the focus (where the receiver is placed).
Find the intercepts of the circle x2 + y2 − 6x + 5 = 0.
Show that the following lines intersect to form a parallelogram. 8x + 10y = 3; 2x − 3y = 5; 4x − 6y = −3; 5y + 4x = 1.
Determine the type of curve for the perimeter of the lighted area on the floor depending on the position of the flashlight and cone as described. See Fig. 21.96.The flashlight is parallel to the
A person exercising on an elliptical trainer is moving her feet along an elliptical path with a horizontal major axis of 32 in. and a vertical minor axis of 10 in. Find the equation of the ellipse if
Find the equation of the hyperbola that has the same foci as the ellipse x2/169 + y2/144 = 1 and passes through (4√2, 3).
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r = er cosθ csc
Identify the curve represented by 4y2 − x2 − 6x − 2y = 14 and view it on a graphing calculator.
Show that the given points are vertices of the given geometric figures.(−5, 6), (0, 8), (−3, 1), and (2, 3) are the vertices of a square.
Plot the given curves in polar coordinates.r = 1 − cos 2θ
Find the polar equation of the line through the polar points (1, 0) and (2, π/2).
The rate of development of heat H (in W) in a resistor of resistance R (in Ω) of an electric circuit is given by H = Ri 2, where i is the current (in A) in the resistor. Sketch the graph of H vs. i,
Find the intercepts of the circle x2 + y2 + 2x − 2y = 0.
Determine the type of curve for the perimeter of the lighted area on the floor depending on the position of the flashlight and cone as described. See Fig. 21.96.The flashlight is directed downward
An Australian football field is elliptical. If a field can be represented by the equationwhat are the dimensions (in m) of the field? x² 1500 + y² 1215 15,
For nonzero values of a, b, and c, find the intercepts of the line ax + by + c = 0.

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