Problem 22 - 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. $$ 9 x^{2}+10 y^{2}=90 $$

Step-by-Step Solution

Answer

Equation in standard form: \(\frac{x^2}{10} + \frac{y^2}{9} = 1\).

Step 1: Identify the Equation

The given equation of the ellipse is \(9x^2 + 10y^2 = 90\). This equation is not in standard form yet.

Step 2: Divide to Normalize

To change the equation into standard form, divide each term by 90: \(\frac{9x^2}{90} + \frac{10y^2}{90} = \frac{90}{90}\). This simplifies to \(\frac{x^2}{10} + \frac{y^2}{9} = 1\).

Step 3: Identify Type and Parameters

The standard form of an ellipse is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) for a horizontal major axis or \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\) for a vertical major axis. Here, \(a^2 = 10\) and \(b^2 = 9\), so \(a = \sqrt{10}\approx 3.16\) and \(b = 3\). The ellipse is horizontal because \(10 > 9\).

Step 4: Determine the Foci

The foci of an ellipse are found using \(c^2 = a^2 - b^2\). Here, \(c^2 = 10 - 9 = 1\), so \(c = \sqrt{1} = 1\). The foci of the ellipse are \((\pm c, 0)\), so \((\pm 1, 0)\).

Step 5: Sketch the Ellipse

To sketch the ellipse, use the center (0,0), vertices \((\pm \sqrt{10}, 0)\), co-vertices \((0, \pm 3)\), and foci \((\pm 1, 0)\). Draw an ellipse that is wider along the x-axis and include these points and the foci in your sketch.

Key Concepts

Standard form of ellipse

When dealing with ellipses, it's crucial to express their equations in their standard form. The standard form helps in understanding the geometric properties easily. An ellipse equation is in standard form when it is written as either \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) for ellipses with a horizontal major axis, or \( \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \) for ellipses with a vertical major axis. To convert an ellipse equation to standard form, all terms need to be adjusted such that the equation equals 1. For instance, starting with the equation \( 9x^2 + 10y^2 = 90 \), dividing every term by 90 results in \( \frac{x^2}{10} + \frac{y^2}{9} = 1 \). This places the ellipse in standard form, identifying the lengths of axes and allowing further analysis.

Sketching ellipses

Once an ellipse's equation is in standard form, sketching it becomes much more straightforward. The key is identifying the orientation and dimensions.

• The center of the ellipse is always at the origin in basic expressions like this one, \((0,0)\).
• To sketch, determine the semi-major and semi-minor axes. These come from \(a = \sqrt{10} \approx 3.16\) and \(b = 3\) in our example.
• Identify the vertices at \((\pm a, 0)\) and co-vertices at \((0, \pm b)\).

Since \(a > b\), the major axis is horizontal, making the ellipse wider along the x-axis. Use these calculated points to draw a smooth, oval shape representing all quadrants equally. Always include the center, vertices, co-vertices, and foci in your sketches for accuracy.

Foci of ellipse

Foci are crucial points within an ellipse that define its shape. Unlike a circle, where all points are equidistant from the center, an ellipse's points are determined based on two foci. The sum of the distances from any point on the ellipse to each focus remains constant. This property gives ellipses their distinct elongated shape.To find the foci of an ellipse, we use the formula \(c^2 = a^2 - b^2\), where \(c\) represents the distance from the center to each focus. From the example, \(c^2 = 10 - 9 = 1\), leading to \(c = \sqrt{1} = 1\). Hence, the foci are positioned at \((\pm c, 0)\) or simply \((\pm 1, 0)\) for our horizontal major axis ellipse.Having the foci included in sketches is essential as it physically conveys the mathematical properties that define the unique nature of ellipses. These points significantly contribute to the ellipse’s definition, affecting its eccentricity and geometry.