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Problem 5.5.27
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.
x = 2 sin t, y = 2 cos t; 0 ≤ t < 2π
- In Exercises 1–8, add or subtract as indicated and write the result in standard form. (7 + 2i) + (1 − 4i)
Problem 1
- In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 3 − 5t, y = 4 + 2t; t = 1
Problem 1
- In Exercises 1–10, perform the indicated operations and write the result in standard form. (8 − 3i) − (17 − 7i)
Problem 1
- In Exercises 1–3, perform the indicated operations and write the result in standard form. (6 − 7i)(2 + 5i)
Problem 1
- In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, 225°)
Problem 1
- In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 7 − 4t, y = 5 + 6t; t = 1
Problem 2
- In Exercises 1–3, perform the indicated operations and write the result in standard form. 5 / 2−i
Problem 2
- In Exercises 1–8, add or subtract as indicated and write the result in standard form. (3 + 2i) − (5 − 7i)
Problem 3
- In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = t² + 1, y = 5 − t³; t = 2
Problem 3
- In Exercises 1–3, perform the indicated operations and write the result in standard form. ___ ___ 2√−49 + 3√−64
Problem 3
- In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, 5π/4)
Problem 3
- In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = t² + 3, y = 6 − t³; t = 2
Problem 4
- In Exercises 1–10, perform the indicated operations and write the result in standard form. (3 − 4i)²
Problem 4
- In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)
Problem 5
- In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2
Problem 5
- In Exercises 1–10, perform the indicated operations and write the result in standard form. (7 + 8i)(7 − 8i)
Problem 5
- In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, π)
Problem 5
Problem 5.15
In Exercises 9–20, find each product and write the result in standard form.
(3 + 5i)(3 − 5i)
Problem 5.27
In Exercises 21–28, divide and express the result in standard form.
2+3i / 2+i
Problem 5.30
In Exercises 29–36, simplify and write the result in standard form.
√−196
Problem 5.33
In Exercises 29–36, simplify and write the result in standard form.
√3² − 4 ⋅ 2 ⋅ 5
Problem 5.43
In Exercises 41–43, eliminate the parameter. Write the resulting equation in standard form.
A hyperbola: x = h + a sec t, y = k + b tan t
Problem 5.45
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Circle: Center: (3,5); Radius: 6
Problem 5.47
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Ellipse: Center: (−2,3); Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center
Problem 5.49
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Hyperbola: Vertices: (4,0) and (−4,0); Foci: (6,0) and (−6,0)
Problem 5.51
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Line: Passes through (−2,4) and (1,7)
Problem 5.57
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
y² = 6x
- In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 2 + 3 cos t, y = 4 + 2 sin t; t = π
Problem 6
- In Exercises 1–10, perform the indicated operations and write the result in standard form. 6 / 5+i
Problem 6