Problem 29 - Intermediate Algebra

Question

Graph equation. \(4(x-1)^{2}+9(y+2)^{2}=36\)

Step-by-Step Solution

Answer

The graph is an ellipse centered at (1, -2) with a major axis of 6 along the x-axis.

Step 1: Identify the Equation Type

The given equation is \(4(x-1)^2 + 9(y+2)^2 = 36\). Notice the form resembles \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), which is the standard form of an ellipse centered at \((h, k)\).

Step 2: Rearrange to Standard Ellipse Form

First, divide each term by 36 to simplify:\[\frac{4(x-1)^2}{36} + \frac{9(y+2)^2}{36} = \frac{36}{36}\]This becomes:\[\frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1\].

Step 3: Identify Ellipse Parameters

From \(\frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1\), we can determine:- The center of the ellipse, \((h, k)\), is \((1, -2)\).- \(a^2 = 9\), so \(a = 3\).- \(b^2 = 4\), so \(b = 2\).

Step 4: Determine Orientation of the Ellipse

Since \(a > b\), the major axis is along the x-direction. This means the ellipse is wider along the x-axis.

Step 5: Write Down the Major and Minor Axis Lengths

The lengths of the major and minor axes are: - Major axis = 2a = 6 units. - Minor axis = 2b = 4 units.

Step 6: Sketch the Ellipse

Start by plotting the center at \((1, -2)\). From this point:- Move 3 units left and right for the major axis (along x-axis).- Move 2 units up and down for the minor axis (along y-axis).Draw an ellipse through these points to complete the sketch.

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse's equation is crucial to understand. It gives us a structured way to work with ellipses. The general standard form of an ellipse centered at \((h, k)\) is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). This equation helps in defining the ellipse's shape, orientation, and dimensions. Here, \((h, k)\) represents the center, \(a\) and \(b\) are the distances from the center to the ellipse along the x and y axes respectively. These distances are critical as they decide the stretching and sizing of the ellipse.

An essential step in working with ellipses is to convert the given equation into this standard form. Just like in the original exercise, we simplify the equation by dividing through by a constant to make it equal to 1. This helps us easily identify the parameters of the ellipse. Once in this form, the characteristics of the ellipse are plainly visible.

Ellipse Parameters

Ellipse parameters provide the necessary details about an ellipse's shape and position in the coordinate plane. In the standard equation \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), the parameters include:

• Center: Located at \((h, k)\).
• Semi-major axis: The term \(a\) represents this and is the longest radius of the ellipse.
• Semi-minor axis: The term \(b\) stands for this shorter radius.
• Axes orientation: If \(a > b\), the major axis is horizontal; if \(b > a\), it's vertical.

In the exercise, identifying these parameters involves breaking down the ellipse's modified equation: \(\frac{(x-1)^2}{9} + \frac{(y+2)^2}{4} = 1\).

From here, the center is given by the values in the expressions \((x-1)\) and \((y+2)\), yielding \((1, -2)\). The semi-axis lengths \(a\) and \(b\) can be derived by taking the square roots of the denominators, giving us \(a = 3\) and \(b = 2\). These parameters reveal not only the size and shape of the ellipse but also its position in the coordinate system.

Major and Minor Axes of an Ellipse

The major and minor axes are fundamental in defining an ellipse. They refer to the longest and shortest diameters of the ellipse, respectively. In any situation where you have an ellipse given in standard form, these axes help you determine the orientation and dimensions of the ellipse.

• Major Axis: The major axis's length is twice that of the semi-major axis, represented as \(2a\).
• Minor Axis: Similarly, the minor axis's length is twice that of the semi-minor axis, noted as \(2b\).

In the example provided, the major axis is determined by the larger value of \(a\) or \(b\). Given \(a = 3 > b = 2\), the major axis is along the x-direction, making it 6 units in total length (\(2 \times 3\)).

Meanwhile, the minor axis stretches 4 units along the y-direction (\(2 \times 2\)).This understanding helps visualize how the ellipse stretches along different axes in the Cartesian plane, painting a more vivid picture of its geometric characteristics.