Problem 36 - Algebra and Trigonometry

Question

\(35-38\) Use a graphing device to graph the conic. $$ 4 x^{2}+9 y^{2}-36 y=0 $$

Step-by-Step Solution

Answer

The conic section is an ellipse centered at (0, 2) with semi-major axis length 2 and semi-minor axis length 3.

Step 1: Rewrite the Equation

First, rewrite the equation \(4x^2 + 9y^2 - 36y = 0\) in a more familiar conic form by isolating the constant on one side. This becomes \(4x^2 + 9(y^2 - 4y) = 0\).

Step 2: Complete the Square for the y-term

Take the y-term \(y^2 - 4y\) and complete the square. Add and subtract \(4\) inside the parentheses: \(y^2 - 4y + 4 - 4 = (y-2)^2 - 4\). The equation now looks like \(4x^2 + 9((y-2)^2 - 4) = 0\).

Step 3: Simplify the Equation

Substitute \((y-2)^2 - 4\) into the equation: \(4x^2 + 9(y-2)^2 - 36 = 0\). Add \(36\) to both sides: \(4x^2 + 9(y-2)^2 = 36\).

Step 4: Divide by the Constant

Divide the entire equation by \(36\) to get \(\frac{4x^2}{36} + \frac{9(y-2)^2}{36} = 1\) or \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\).

Step 5: Identify the Conic Section

The final form \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\) is recognizable as an ellipse centered at \((0, 2)\). The semi-major axis in the y-direction has a length of \(2\) and the semi-minor axis in the x-direction has a length of \(3\).

Step 6: Graph the Equation

Using a graphing device, plot the ellipse with the center at \((0, 2)\). Draw the semi-major axis along the vertical from \(y = 0\) to \(y = 4\) and the semi-minor axis along the horizontal from \(x = -3\) to \(x = 3\). Ensure the shape is symmetrical about the center.

Key Concepts

Ellipse

An ellipse is one of the four types of conic sections, which are curves obtained by intersecting a cone with a plane. Ellipses appear in various natural and scientific contexts, from the orbits of planets to the shape of certain lenses. In mathematics, an ellipse is a set of points where the sum of the distances from two fixed points, called foci, is constant.

An ellipse is defined by its general equation:

• If \(a > b\), the ellipse is oriented horizontally, and the equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
• If \(b > a\), the ellipse is oriented vertically, and the equation is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\).

The axis with a larger denominator is the major axis, determining the ellipse's shape. In our example, \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\), the major axis is along the x-direction. These key characteristics help identify and graph ellipses quickly.

Completing the Square

Completing the square is a fundamental algebraic technique used to transform a quadratic equation into a perfect square trinomial. This method is especially useful for rewriting conic section equations in their standard forms, which makes identifying the type of conic easier.

Here's how it works when applied to a quadratic term, for instance in the example \(y^2 - 4y\):

• Identify the coefficient of the linear term, which is \(4\), halve it to get \(2\), and square it to obtain \(4\).
• Add and subtract this squared term inside the parentheses resulting in \(y^2 - 4y + 4 - 4\).
• This transforms into \( (y-2)^2 - 4\).

Completing the square not only simplifies the equation but also reveals the conic's center, in this case shifting the center along the y-axis. This method streamlines solving and graphing conic equations significantly.

Graphing Conic Sections

Graphing conic sections, like ellipses, involves recognizing the type of conic and plotting it correctly on the coordinate plane. Each conic section—circle, ellipse, parabola, or hyperbola—has a distinct shape and equation.

For an ellipse, the equation tells us:

• The center of the ellipse, derived from the terms \(h,k\) in the equation \((x-h)^2 + (y-k)^2\).
• The lengths of the axes, with the larger denominator indicating the direction of the major axis.

To graph the example \(\frac{x^2}{9} + \frac{(y-2)^2}{4} = 1\):

• The center is at \(0,2\).
• The x-axis (horizontal axis) extends \(3\) units on either side of the center, and the y-axis (vertical axis) extends \(2\) units up and down from the center.
• Plot these points and draw a smooth, symmetrical curve around the center to make the ellipse.

Graphing not only helps visualize the equation but also provides insights into the conic's properties and how they relate to its real-world applications.