Problem 91 - A Graphical Approach to College Algebra

Question

Solve each problem. A patient's kidney stone is placed 12 units away from the source of the shock waves of a lithotripter. The lithotripter is based on an ellipse with a minor axis that measures 16 units. Find an equation of an ellipse that would satisfy this situation.

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{100} + \frac{y^2}{64} = 1\).

Step 1: Understanding the Problem

The problem involves an ellipse used in a lithotripter machine, which has two foci. The lithotripter sends shock waves from one focus that are reflected from the surface of the ellipse and converge at the other focus (where the kidney stone is located). We know the distance from the focus to the ellipse (12 units) and the length of the minor axis (16 units).

Step 2: Identify Parameters of the Ellipse

The equation of an ellipse with a horizontal major axis is given by: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]where - \(a\) is the semi-major axis, - \(b = 8\) (half of the minor axis), - \(c\) is the distance from the center to each focus. Given that \(2c = 12\), we find \(c = 6\).

Step 3: Calculate the Semi-Major Axis

Use the relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the distance to the foci \(c\): \[ c^2 = a^2 - b^2 \]Substituting \(b = 8\) and \(c = 6\):\[ 6^2 = a^2 - 8^2 \]\[ 36 = a^2 - 64 \]\[ a^2 = 100 \]\[ a = 10 \]

Step 4: Write the Equation of the Ellipse

With \(a = 10\), \(b = 8\), and assuming the center of the ellipse is at the origin (\((h, k) = (0, 0)\)), the equation of the ellipse is:\[ \frac{x^2}{10^2} + \frac{y^2}{8^2} = 1 \]which simplifies to:\[ \frac{x^2}{100} + \frac{y^2}{64} = 1 \]

Step 5: Verify the Solution

The derived equation \(\frac{x^2}{100} + \frac{y^2}{64} = 1\) matches the problem requirements using the provided distances for the minor axis and the focal points. Double-checking calculations confirms the correct values for \(a\) and \(b\).

Key Concepts

Lithotripter

A lithotripter is a medical device used to break down kidney stones within the body. It employs focused shock waves, which converge at the stone, shattering it into small, easily passable pieces. The principle behind the lithotripter is based on the properties of an ellipse. An ellipse is a geometric shape with two focal points. In the lithotripter:

• One focus is where the shock waves originate.
• The other focus is where the kidney stone is located.

The waves reflect off the surface of the elliptical path and concentrate at the second focus, targeting the stone. This helps in treating kidney stones non-invasively and effectively.

Semi-Major Axis

The semi-major axis is a key characteristic of an ellipse. It is the longest radius, stretching from the center to one endpoint of the ellipse's major axis. This concept is crucial because it represents half the maximum width of the ellipse. For an ellipse:

• The length of the entire major axis is twice that of the semi-major axis.
• It serves as a baseline for determining the size and shape of the ellipse.

Knowing the semi-major axis is essential when calculating the ellipse's equation. In our example, the semi-major axis is determined to be 10 units. This represents half the distance across the longest part of the ellipse, which helps in accurately shaping the device's shock path.

Semi-Minor Axis

The semi-minor axis of an ellipse is the shortest radius stretching from the center to the ellipse's edge, along its minor axis. It contributes to defining the shape and dimensions of the ellipse, alongside the semi-major axis. Important points about the semi-minor axis include:

• It is always perpendicular to the semi-major axis.
• The full minor axis is double the length of the semi-minor axis.

In the given problem, the semi-minor axis is 8 units, indicating the short span of the elliptical path. This aspect is important for a lithotripter as it shapes how the shock waves reflect and converge at the desired point.

Foci of an Ellipse

The foci (plural of focus) are two distinct points located on the major axis of the ellipse. Their significance lies in their role in the reflective properties of the ellipse. Key elements of the foci include:

• The total distance from one focus to any point on the ellipse and back to the other focus remains constant.
• They dictate the trajectory of paths, like shock waves in a lithotripter.

For the lithotripter, the kidney stone is placed at one focus, while the shock wave source is at the other. In the problem, the distance between the foci is vital, calculated using the equation derived from the ellipse properties. Here, with a center at the origin, each focus is 6 units away, ensuring the ellipse efficiently directs waves to the stone.