Problem 36 - Algebra and Trigonometry

Question

Find an equation for the ellipse that satisfies the given conditions. Endpoints of minor axis \((0, \pm 3),\) distance between foci 8

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).

Step 1: Identify ellipse properties from minor axis endpoints

The endpoints of the minor axis, located at \((0, \pm 3)\), indicate that the minor axis is vertical with the center of the ellipse at \((h, k) = (0, 0)\) and half-length of the minor axis \(b = 3\).

Step 2: Calculate the semi-major axis and its orientation

The distance between the foci is given as 8, which means \(2c = 8\), so \(c = 4\). Since the minor axis is vertical, the major axis is horizontal, making \(a\) the semi-major axis and \(c^2 = a^2 - b^2\).

Step 3: Apply the relationship between a, b, and c

Using the equation \(c^2 = a^2 - b^2\) with \(b = 3\) and \(c = 4\), calculate \(a^2\):\[4^2 = a^2 - 3^2\]\[16 = a^2 - 9\]Add 9 to both sides:\[a^2 = 25\]Thus, \(a = 5\).

Step 4: Formulate the equation of the ellipse

The general equation for a horizontally oriented ellipse with center at \((h, k)\) is:\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]Substitute the identified values (\(h = 0\), \(k = 0\), \(a = 5\), \(b = 3\)):\[\frac{x^2}{25} + \frac{y^2}{9} = 1\]

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone, and the resulting shapes are geometric figures called: parabolas, ellipses, circles, and hyperbolas. Depending on the angle and position at which the plane slices through the cone, different sections are produced. An ellipse, for example, is formed when a plane cuts through the cone at an angle such that it does not intersect the base. This makes ellipses one of the key shapes studied in conic sections. Understanding conic sections gives great insight into the geometric properties and equations that describe these shapes in plane geometry.

Minor and Major Axes

The major and minor axes of an ellipse are two significant line segments that pass through its center at distinct angles. The major axis is the longest diameter of the ellipse, which runs through its foci. The endpoints of the major axis correspond to the widest span of the ellipse. In contrast, the minor axis is the shortest diameter and is perpendicular to, and bisected by, the major axis.

In the context of our problem, the endpoints of the minor axis are given as \(0, \pm 3\), indicating it is vertical. The minor axis has a half-length, often called "b," which is 3 in this case. Since the minor axis is vertical, the major axis, with half-length "a,"] lies horizontally and its calculation is central to deriving the ellipse's equation.

Foci of an Ellipse

Ellipses have two specific internal points called foci. The sum of the distances from any point on the ellipse to these two foci is constant. This property differentiates an ellipse from other conic sections. The foci are located along the major axis, equidistant from the ellipse’s center.

In the problem, the distance between the foci is provided as 8, which signifies that 2c = 8, leading to c = 4. This distance helps determine the orientation and dimensions of the ellipse using the relationship \(c^2 = a^2 - b^2\), and is a critical component in defining the spatial configuration of the ellipse within a plane.

Equation of an Ellipse

The equation of an ellipse is an expression that encapsulates its geometric properties in algebraic form. The general equation for an ellipse centered at \(h, k\) is:

• Horizontal major axis: \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\)
• Vertical major axis: \(\frac{(y - k)^2}{a^2} + \frac{(x - h)^2}{b^2} = 1\)

The semi-major axis and semi-minor axis measurements, "a" and "b," play crucial roles. Given the center at \(0, 0\) and minor axis length, along with the distance between the foci, this information assists in identifying whether the ellipse’s major axis is horizontal or vertical, thus influencing which form of the equation to use. Since our ellipse has a horizontal major axis, we used \(h = 0, k = 0, a = 5, b = 3\) in the equation, resulting in: \(\frac{x^2}{25} + \frac{y^2}{9} = 1\).