Problem 19 - Thomas Calculus

Question

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 2 x^{2}+y^{2}=2 $$

Step-by-Step Solution

Answer

The ellipse is in standard form: \(\frac{x^2}{1} + \frac{y^2}{2} = 1\), with foci at \((0, \pm 1)\).

Step 1: Write the Equation in Standard Form

The given equation of the ellipse is \(2x^2 + y^2 = 2\). To write it in standard form, we need to have the right-hand side equal to 1. To do this, divide the entire equation by 2. This gives us: \[ \frac{x^2}{1} + \frac{y^2}{2} = 1 \] This equation is now in the form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Step 2: Identify Ellipse Parameters

From the standard form equation \(\frac{x^2}{1} + \frac{y^2}{2} = 1\), identify the values: - \(a^2 = 1\), so \(a = 1\) - \(b^2 = 2\), so \(b = \sqrt{2}\) Here, since \(b > a\), the ellipse is vertical.

Step 3: Find the Foci of the Ellipse

For an ellipse where \(b > a\), the distance to the foci, \(c\), is found using \(c = \sqrt{b^2 - a^2}\). With \(a = 1\) and \(b = \sqrt{2}\), we have: \[ c = \sqrt{2 - 1} = 1 \] So, the foci are located at \((0, \pm 1)\).

Step 4: Sketch the Ellipse

Draw the ellipse centered at the origin \((0,0)\) with a vertical major axis. - The major semi-axis length \(b = \sqrt{2}\), so it extends \(\pm \sqrt{2}\) units above and below the origin on the y-axis.- The minor semi-axis length \(a = 1\), so it extends \(\pm 1\) units left and right on the x-axis.- Mark the foci at \((0, 1)\) and \((0, -1)\).

Key Concepts

Standard Form of Ellipse

To express an ellipse equation in its standard form, it must fit the template \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This template is pivotal as it allows you to directly identify the properties of the ellipse, such as the major and minor axes. Begin by ensuring the right side of your equation equals 1. In our exercise, the original equation is \( 2x^2 + y^2 = 2 \). To transform it, divide every term by 2, resulting in \( \frac{x^2}{1} + \frac{y^2}{2} = 1 \). This now follows the standard form. Here, the coefficients underneath \(x^2\) and \(y^2\) are effectively \(a^2\) and \(b^2\) respectively.

Ellipse Parameters

Identifying the parameters of an ellipse involves determining the lengths of its axes based on the standard form equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). From this format, the parameters, \(a\) and \(b\), inform the size of the ellipse's axes.

• Here, \(a^2 = 1\), thus \(a = 1\).
• For \(b^2 = 2\), \(b = \sqrt{2}\).

As the larger value \(b\) is associated with the \(y\)-term, this ellipse is vertical, signifying a longer vertical axis compared to the horizontal one. These parameters are crucial as they dictate the overall shape and orientation of the ellipse.

Foci of an Ellipse

The foci of an ellipse are two special points along the major axis. They play a key role in defining the geometric properties of an ellipse. Determining their position involves using the calculated values for \(a\) and \(b\). When \(b > a\) in a vertical ellipse, use the formula for the foci distance, \(c = \sqrt{b^2 - a^2}\).

• From the earlier calculation, \(a = 1\) and \(b = \sqrt{2}\).
• Calculate \(c\) as \(\sqrt{2 - 1} = 1\).

Thus, the foci are located at \((0, \pm 1)\). These points are symmetric about the center of the ellipse at the origin \((0, 0)\). Understanding the location of foci is important as they highlight the unique property of ellipses: the sum of distances from any point on the ellipse to the two foci is constant.

Graphing Ellipses

Graphing an ellipse involves plotting its shape based on its parameters and foci. Begin by marking the center of the ellipse, which in this case is at the origin \((0, 0)\). From there, use the lengths \(a\) and \(b\) to determine the extremities of the ellipse.

• The major semi-axis length, \(b = \sqrt{2}\), defines the vertical stretch, extending \(\pm \sqrt{2}\) units from the center on the \(y\)-axis.
• The minor semi-axis length, \(a = 1\), describes the horizontal reach, stretching \(\pm 1\) unit on the \(x\)-axis.

Identify and plot the foci at \((0, 1)\) and \((0, -1)\), enhancing the visualization of the ellipse's geometric properties. In combination, these steps provide a complete graph of the ellipse, depicting its full scale and positioning with clarity.