Problem 19 - Algebra 2

Question

Find the foci for each equation of an ellipse. Then graph the ellipse. $$ \frac{x^{2}}{9}+\frac{y^{2}}{25}=1 $$

Step-by-Step Solution

Answer

The foci of the ellipse are at points (0, 4) and (0, -4). The graph of the ellipse would pass through the points (3, 0) , (-3, 0) , (0, 5) , and (0, -5) with foci at points above and below the origin.

Step 1: Identify the semimajor and semiminor axes

Looking at the given equation, identify the values for \(a\) and \(b\). The given ellipse equation is in standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Therefore, \(a\) is the square root of denominator of \(y^2\) term (largest denominator) and \(b\) is the square root of the denominator of \(x^2\) term. Hence, \(a = 5\) and \(b = 3\).

Step 2: Calculate the foci

Substitute the determined \(a\) and \(b\) values in the foci formula \(c = \sqrt{a^2 - b^2}\). So, \(c = \sqrt{5^2 - 3^2} = \sqrt{16} = 4\). The coordinates for the foci would be (0, 4) and (0, -4) since there is a larger denominator under the \(y^2\) term, meaning we're stretching more along the y-axis.

Step 3: Sketch the Ellipse

Draw the x and y axes. Plot the points (0, 5) and (0, -5) as well as (3, 0) and (-3, 0) to serve as the vertices and co-vertices respectively. Similarly, plot the foci determined in Step 2. Then sketch the ellipse using these points.

Key Concepts

Foci of an Ellipse

Understanding the foci of an ellipse helps to reveal the inherent symmetry and balance of this geometric shape. In any ellipse, there are two fixed points known as the foci (singular: focus). While these might seem complex at first, they are actually quite simple to pinpoint after understanding the underlying principles. To find the foci of an ellipse, the formula used is:

• If the equation is of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and \(b > a\), then the foci are located along the y-axis at \((0, \pm c)\).
• The formula to calculate \(c\) is \(c = \sqrt{b^2 - a^2}\).

Given our original equation \( \frac{x^2}{9} + \frac{y^2}{25} = 1 \), it follows that \( a = 3 \), \( b = 5 \), and therefore, \( c = \sqrt{5^2 - 3^2} = 4 \). Consequently, the foci are \((0, 4)\) and \((0, -4)\). The foci's position along the y-axis arises due to the larger \( a^2 \) corresponding to the \( y^2 \) term, aligning along the vertical axis of the ellipse.

Graphing Ellipses

Visualizing an ellipse through graphing provides invaluable insight into its structure. Firstly, identify the semi-major and semi-minor axes. The semi-major axis is tied to the larger denominator under the variables, and the semi-minor to the smaller one.

• The length of the semi-major axis is \( 2a \) and of the semi-minor axis is \( 2b \).

For our equation \( \frac{x^2}{9} + \frac{y^2}{25} = 1\), the semi-major axis runs along the y-axis with \(a = 5\), and the semi-minor axis along the x-axis with \(b = 3\). To graph it:

• Plot the vertices at (0, 5) and (0, -5).
• Plot the co-vertices at (3, 0) and (-3, 0).

Drawing a smooth, oval shape through these points, with the foci (0, 4) and (0, -4) inside, completes the ellipse. Graphing serves as a tangible verification of our calculations and strengthens our geometric understanding.

Conic Sections

Ellipses are part of a family of curves known as conic sections, which arise from intersecting a plane with a double-napped cone. Conic sections include ellipses, parabolas, and hyperbolas, each defined by the angle of intersection between the cone and the plane.

• If the plane cuts parallel to the base of the cone, we get a circle.
• If the plane intersects at a smaller angle than the parabola but not steep enough for a hyperbola, an ellipse forms.
• Changes in this angle or distance help determine whether conic sections distort into eccentric shapes.

Ellipses are paramount in various scientific fields because of their natural occurrence, such as planetary orbits. Fundamental to their definition is the relationship between distances to two focus points, ensuring that the sum remains constant throughout. This balance of equidistant points relative to the ellipse's curve is integral to its consistent geometrical properties. Understanding conics offers both aesthetic and practical insights into the mathematical world.