For parametric equations x=a cos t and y=b sin t, describe how the values of a and b determine which conic section will be traced.
The parametric equations x=a cos t and y=b sin t represent a curve in the xy-plane. The values of a and b play a crucial role in determining which type of conic section is traced by this curve.
1. If a = b, the curve traced will be a circle. Both x and y coordinates will vary in such a way that the resulting curve will form a circle centered at the origin.
2. If a ≠ b, the curve traced will be an ellipse. The major axis of the ellipse will align with the x-axis, and the minor axis will align with the y-axis. The lengths of the axes will be determined by the values of a and b.
3. If a > b, the major axis of the ellipse will be along the x-axis, and the minor axis will be along the y-axis. The ellipse will be stretched in the x-direction.
4. If b > a, the major axis of the ellipse will be along the y-axis, and the minor axis will be along the x-axis. The ellipse will be stretched in the y-direction.
In summary, the values of a and b determine whether the curve traced will be a circle or an ellipse, as well as the orientation and extent of stretching along the x and y axes for the case of an ellipse.
To understand how the values of a and b determine the conic section traced by the parametric equations x = a cos(t) and y = b sin(t), let's break it down step by step:
1. Parametric Equations: Parametric equations represent the values of x and y in terms of a parameter, in this case, t.
2. Form of Conic Section: The conic section can be identified by the general equation that defines it. The most common forms are:
- Ellipse: (x² / A²) + (y² / B²) = 1, where A and B are positive constants.
- Circle: (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is the radius.
- Parabola: y = ax² + bx + c, where a, b, and c are constants.
- Hyperbola: (x² / A²) - (y² / B²) = 1 or (y² / B²) - (x² / A²) = 1, where A and B are positive constants.
3. Parameter t: Since x and y are expressed in terms of t, the parameter t represents the angle at which the point (x, y) is positioned relative to the origin (0, 0). As t varies, the point moves along the curve.
4. Identifying the Conic Section:
- If a = b, the curve traced by the parametric equations will be a circle.
- If a > b or a < b, the curve traced will be an ellipse.
- If a = 0 or b = 0, the curve traced will be a straight line.
- If a ≠ 0 and b ≠ 0, the curve traced will be a hyperbola.
- If a and b have the same sign, the hyperbola will have a vertical orientation.
- If a and b have opposite signs, the hyperbola will have a horizontal orientation.
Therefore, by examining the values of a and b, you can determine which conic section will be traced by the parametric equations x = a cos(t) and y = b sin(t).