Problem 38 - College Trigonometry

Question

In Exercises \(25-39\), find a parametric description for the given oriented curve. the ellipse \(9 x^{2}+4 y^{2}+24 y=0\), oriented clockwise (Shift the parameter so \(t=0\) corresponds to \((0,0) .)\)

Step-by-Step Solution

Answer

The parametric equations are \(x = 2\sin(t)\), \(y = -3 - 3\cos(t)\) for clockwise orientation starting at \((0,0)\).

Step 1: Rewrite the Ellipse Equation

The ellipse equation is given as \(9x^2 + 4y^2 + 24y = 0\). First, complete the square for the \(y\) terms to put the equation in a more recognizable form. Factor \(4\) out of the \(y\) terms: \(9x^2 + 4(y^2 + 6y) = 0\). Complete the square inside the parenthesis for \(y^2 + 6y\): \(y^2 + 6y + 9 - 9 = (y+3)^2 - 9\). Substitute back: \(9x^2 + 4((y+3)^2 - 9) = 0\). Simplify: \(9x^2 + 4(y+3)^2 - 36 = 0\). Rearrange: \(9x^2 + 4(y+3)^2 = 36\).

Step 2: Divide to Normalize the Ellipse Equation

Divide the whole equation by 36 to match the standard form of an ellipse. This results in \(\frac{x^2}{4} + \frac{(y+3)^2}{9} = 1\). This is the equation of an ellipse centered at \((0,-3)\) with semi-major axis along \(y\) and semi-minor axis along \(x\).

Step 3: Establish Parametric Equations with Clockwise Orientation

The standard parametric form for an ellipse centered at \((h,k)\) is \(x = h + a \cos(t)\) and \(y = k + b \sin(t)\), where \(a\) and \(b\) are the semi-axes lengths. Here, \(a = 2\) and \(b = 3\), giving: \(x = 2\cos(t)\) and \(y = -3 + 3\sin(t)\). For clockwise orientation, modify the parametrization: \(x = 2\cos(t)\) and \(y = -3 - 3\sin(t)\).

Step 4: Shift Parameter for Initial Condition \((0,0)\)

To ensure \(t = 0\) corresponds to the point \((0,0)\), set \(x = 2\cos(t) = 0\) which implies \(t = \frac{\pi}{2}\). Adjust the parameter \(t\) by subtracting \(\frac{\pi}{2}\): redefine \(t' = t + \frac{\pi}{2}\). The new parametric equations become \(x = 2\cos(t')\) and \(y = -3 - 3\sin(t')\) with \(t' = 0\) at \((0,0)\). Use \(t\) for simplicity: \(x = 2\sin(t)\), \(y = -3 - 3\cos(t)\), ensuring \((x, y) = (0, 0)\) when \(t = 0\).

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that resembles a flattened circle. It has two main axes: the major axis and the minor axis which are perpendicular to each other. In this problem, we started with the equation \( 9x^2 + 4y^2 + 24y = 0 \). To understand the shape, we rearranged it using algebraic techniques, such as completing the square, to achieve

• a form where \( \frac{x^2}{4} + \frac{(y+3)^2}{9} = 1 \).
• This identifies the ellipse as centered at \((0, -3)\)
• Semi-major axis (longest) along the y-axis with length 3 and the semi-minor axis (shortest) along the x-axis with length 2.

The ellipse's geometry meaningfully represents a set of all points, where the sum of distances from two fixed points (foci) is constant. This exercise embodies geometry and algebra in its manipulation and interpretation of the initial equation.

Orientation

Orientation is an important concept related to the direction in which a curve is traced by a moving point. For this ellipse problem, the orientation specifies that the curve should be traced clockwise. Initially, we obtained parametric equations from the standard form: \( x = 2\cos(t) \) and \( y = -3 + 3\sin(t) \). To orient the path clockwise:

• We changed the parametric description to \( x = 2\cos(t) \) and \( y = -3 - 3\sin(t) \).
• This adjustment effectively reverses the direction of the `y` path, hence switching the orbit from the usual counterclockwise to clockwise.

Understanding orientation helps ensure that the physical representation—whether it's paths taken by satellites or animations in graphics—matches the desired directional flow.

Trigonometric Functions

Trigonometric functions such as sine and cosine are crucial for describing periodic phenomena. In this context, they allow a precise description of the ellipse in terms of position over time. Given

• \( x = 2\sin(t) \)
• \( y = -3 - 3\cos(t) \)

these functions oscillate between -1 and 1, naturally generating the smooth path of an ellipse.Sine and cosine effectively create circular and elliptical paths due to their periodic nature. Cosine represents horizontal movement while sine denotes vertical movement. These functions ensure that as t varies, all points on the ellipse are accounted for in a steady, repeatable manner.

Coordinate Transformation

Coordinate transformation is crucial in geometry, particularly when converting between different forms of representing a shape, such as from an implicit equation to parametric form. Initially, the ellipse's description was in its standard quadratic form. Through completing the square and normalization:

• Transformed into \( \frac{x^2}{4} + \frac{(y+3)^2}{9} = 1 \),

which reveals its positioning and orientation.
Afterward, parametric equations were derived:

• \( x = 2\sin(t) \)
• \( y = -3 - 3\cos(t) \)

These transformations allow us to map coordinates in a way that aligns them with specific needs, like

• ensuring a clockwise motion,
• tailoring the start point of \(t = 0 \) to be at \( (0, 0) \).

Ultimately, this process shows flexibility in mathematical representations, offering varied perspectives for analysis and application.