Problem 34 - Algebra and Trigonometry

Question

Find an equation for the ellipse that satisfies the given conditions. Foci: \((0, \pm 3),\) vertices: \((0, \pm 5)\)

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

Step 1: Identify the Center

The foci are located at \((0, \pm 3)\), and the vertices are at \((0, \pm 5)\). Since both foci and vertices are aligned along the \(y\)-axis and centered at \((0,0)\), the center of the ellipse is \((0,0)\).

Step 2: Determine the Values of a and c

The vertices \((0, \pm 5)\) indicate that \(a = 5\). The foci \((0, \pm 3)\) indicate that \(c = 3\). In an ellipse, \(c^2 = a^2 - b^2\).

Step 3: Calculate the Value of b

Using the formula \(c^2 = a^2 - b^2\), we substitute the known values: \(3^2 = 5^2 - b^2\). This simplifies to \(9 = 25 - b^2\). Solving for \(b^2\), we get \(b^2 = 16\). Thus, \(b = 4\).

Step 4: Write the Equation of the Ellipse

With \(a = 5\), \(b = 4\), and the center at \((0,0)\), the equation of the ellipse is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), which becomes \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

Key Concepts

Vertices of an Ellipse

The vertices of an ellipse are crucial points that define its overall shape and orientation. These vertices are the most distant points from the center along each of the ellipse's principal axes. In our given exercise, the ellipse has its vertices at \((0, \pm 5)\). Here, we notice that the vertices are aligned vertically along the \(y\)-axis.

• The vertices directly indicate the value of \(a\).
• In this scenario, \(a = 5\) because the vertices are 5 units away from the center \((0,0)\).

Understanding the position of vertices helps in sketching the ellipse and forms a base for determining other parameters, such as the foci and the length of the semi-minor axis.

Foci of an Ellipse

The foci of an ellipse are two distinct points situated along the ellipse's principal axis. An ellipse is defined such that the sum of the distances from any point on the ellipse to these two foci is constant. In our example, the foci are given as \((0, \pm 3)\), which also lie along the \(y\)-axis.

• The distance from the center \((0,0)\) to each focus is \(c = 3\).
• The relationship \(c^2 = a^2 - b^2\) helps in finding the semi-minor axis of the ellipse.

Understanding the location of the foci is essential in the derivation of the ellipse's equation and is a critical aspect of its geometric properties.

Center of an Ellipse

The center of an ellipse serves as the midpoint from which both the axes and various components radiate. It is the balance point of the ellipse, from which both the vertices and foci are equidistantly distributed in opposite directions. For this exercise, both the foci \((0, \pm 3)\) and vertices \((0, \pm 5)\) are aligned along the \(y\)-axis, centering the ellipse at \((0,0)\).

• The center coordinates are crucial for setting the ellipse's equation in its standard form.
• Changes in the center position directly affect the translation of the entire ellipse, without altering its shape.

Identifying the center is fundamental as it ensures that all measurements from the center, especially those for "a" and "c," are accurately determined.

Standard Form of Ellipse Equation

The standard form of an ellipse equation is essential for expressing an ellipse algebraically and is commonly written as \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) for a vertical ellipse. It integrates all the distances calculated along the ellipse's axes. Based on our exercise:

• Since the vertices are along the \(y\)-axis, it suggests the vertical orientation of the ellipse.

With \(a = 5\) and \(b = 4\), the equation becomes \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

• This equation encapsulates all the elements of the ellipse: center, vertices, and foci.
• It is crucial for graphing the ellipse and understanding its symmetrical properties.

Familiarity with this standard form allows for easier manipulation and recognition of ellipses in various mathematical contexts.