Question

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle, \quad\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is a circle? b. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is an ellipse?

Short answer

For the path to form a circle, the conditions are: - (a - h)² + (c - k)² = r² - b² + d² = r² For the path to form an ellipse, the conditions are: - c² + d² = a² - a² + b² = b²

Step-by-step solution

Write down the general equation for a circle and an ellipse

The general equation for a circle is: \((x - h)^2 + (y - k)^2 = r^2\) The general equation for an ellipse is: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\)

Substitute the expressions for x and y from the given path

For the given path \(\mathbf{r}(t) = \langle a \cos t + b \sin t, c \cos t + d \sin t \rangle\), substitute \(x = a \cos t + b \sin t\) and \(y = c \cos t + d \sin t\) in the circle and ellipse equations. Circle: \([(a \cos t + b \sin t) - h]^2 + [(c \cos t + d \sin t) - k]^2 = r^2\) Ellipse: \(\frac{[(a \cos t + b \sin t) - h]^2}{a^2} + \frac{[(c \cos t + d \sin t) - k]^2}{b^2} = 1\)

Simplify the equations and find conditions for circle and ellipse

For the path to form a circle: Observe that in a circle with \(r^2\), the coefficients in front of \(\cos^2 t\) and \(\sin^2 t\) in x and y coordinates must be the same and their centers must be the same. \([(a - h) \cos t + b \sin t]^2 + [(c - k) \cos t + d \sin t]^2 = r^2\) Clearly, we must have: 1. \((a - h)^2 + (c - k)^2 = r^2\) 2. \(b^2 + d^2 = r^2\) The conditions for the path to form a circle are: - \((a - h)^2 + (c - k)^2 = r^2\) - \(b^2 + d^2 = r^2\) For the path to form an ellipse: \(\frac{[(a - h) \cos t + b \sin t]^2}{a^2} + \frac{[(c - k) \cos t + d \sin t]^2}{b^2} = 1\) The centers \((h, k)\) must be inside the ellipse, and the coefficients in front of \(\cos^2 t\) and \(\sin^2 t\) for x and y coordinates must be the same for a and b, respectively. Clearly, we must have: \(c^2 + d^2 = a^2\) and \(a^2 + b^2 = b^2\) The conditions for the path to form an ellipse are: - \(c^2 + d^2 = a^2\) - \(a^2 + b^2 = b^2\) In summary: For the path to be a circle, the conditions are: - \((a - h)^2 + (c - k)^2 = r^2\) - \(b^2 + d^2 = r^2\) For the path to be an ellipse, the conditions are: - \(c^2 + d^2 = a^2\) - \(a^2 + b^2 = b^2\)

Key concepts

Circle Parametric Equations

Parametric equations offer a powerful way to describe geometrical shapes, such as circles, without relying on traditional x-y coordinates. When we're dealing with a circle, the parametric equations use trigonometric functions to define each coordinate independently in terms of a single parameter, usually denoted as 't', which often represents time.

When an object is moving along a circular path, its position at any time 't' can be described using the equations. For a circle centered at (h, k) with radius 'r', the parametric equations are:

\( x(t) = h + r \times \text{cos}(t) \)
\( y(t) = k + r \times \text{sin}(t) \)

These equations ensure that for every value of 't', the point \((x(t), y(t))\) will lie on the circle. To ensure that the path is indeed circular using the given vector function \( \mathbf{r}(t) \), the coefficients of \(\text{cos}^2(t)\) and \(\text{sin}^2(t)\) after squaring and combining like terms must be equal, reflecting the uniform radius 'r' for all points on the circle.

Ellipse Parametric Equations

An ellipse, which is a circle stretched along the x or y-axis, can also be described using parametric equations. For an ellipse centered at (h, k) with semi-major axis 'a' along the x-axis and semi-minor axis 'b' along the y-axis, the parametric equations are:

\( x(t) = h + a \times \text{cos}(t) \)
\( y(t) = k + b \times \text{sin}(t) \)

These equations allow us to trace the path of an object moving along an elliptical route. In the context of the given exercise, conditions for the coefficients (a, b, c, d) of the vector function \( \mathbf{r}(t) \) to represent an ellipse include distinct coefficients for \(\text{cos}^2(t)\) and \(\text{sin}^2(t)\) that correspond to the squared ratios of the ellipse's axes. Therefore, the parameter 't' scales the cosine and sine functions by different amounts, stretching the circle into an ellipse.

Vector Functions

Vector functions are an essential tool in multivariable calculus, providing a way to connect algebra and geometry via vectors. They describe the movement of an object in a space of any number of dimensions, assigning a vector to each scalar parameter 't'.

The given vector function \( \mathbf{r}(t) = \langle a \text{cos}(t) + b \text{sin}(t), c \text{cos}(t) + d \text{sin}(t) \rangle \) is the algebraic representation of the object's path in two dimensions, where 'a', 'b', 'c', and 'd' are scalar coefficients and 't' varies over time

.In this context, the components of \(\mathbf{r}(t)\) are built from the basic trigonometric functions, and either describe a circle or an ellipse based on the specific conditions detailed in the steps of the problem solution.

Trigonometry in Calculus

Trigonometry is fundamental in calculus, especially when dealing with periodic functions and motions like those of the circle and ellipse described in the given problem. Trigonometric functions, which are the sine and cosine functions in this case, are used to model oscillations and rotational movements.

Using calculus, we can differentiate and integrate these trigonometric functions to find properties such as velocity, acceleration, and area under the curve. In the case of the described circular and elliptical paths, trigonometry provides the framework for expressing the coordinates as functions of 't', enabling us to solve complex problems in kinematics and other fields that involve cyclical or rotational motion.