Problem 48 - Precalculus: Mathematics for Calculus

Question

Find an equation for the hyperbola that satisfies the given conditions. Foci: \((\pm \sqrt{10}, 0),\) hyperbola passes through \((4, \sqrt{18})\)

Step-by-Step Solution

Answer

The hyperbola is \(\frac{x^2}{5} - \frac{y^2}{5} = 1\).

Step 1: Understand the Standard Equation of a Hyperbola

The standard equation of a hyperbola centered at the origin with horizontal transverse axis is given by: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] where \(a\) is the distance from the center to each vertex and \(c\) is the distance from the center to each focus. The relationship between \(a\), \(b\), and \(c\) is given by \(c^2 = a^2 + b^2\).

Step 2: Identify c from the Foci

The given foci are \((\pm \sqrt{10}, 0)\). Since the foci of the hyperbola are \((\pm c, 0)\), we have \(c = \sqrt{10}\).

Step 3: Use the Given Point to Find the Equation Parameters

We substitute \(x = 4\) and \(y = \sqrt{18}\) into the standard hyperbola equation. Use these coordinates along with known \(c\) to solve for \(a^2\) and \(b^2\).

Step 4: Establish Relationship between c, a, and b

Using \(c^2 = a^2 + b^2\), replace \(c^2 = 10\) in the equation. Then use the previous step to solve the equation \(10 = a^2 + b^2\) after expressing \(b^2\) in terms of \(a^2\) from the point substitution results, solving for \(a^2\) and \(b^2\).

Step 5: Substitute and Solve

Substitute all values found back into \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) to check against the respective values. Conclude with a full solution in terms of \(x\) and \(y\). Determine the necessary \(a^2\) and verify against point \((4, \sqrt{18})\).

Key Concepts

Foci of the Hyperbola

The foci are crucial elements of a hyperbola. They lie along the transverse axis and are equidistant from the center of the hyperbola. For a hyperbola centered at the origin, you'll have foci located at

• defining points on the x-axis as defining points at ( ±c ,0). In this exercise, the provided foci are at ( ± √ 10,0), which tells us the hyperbola is horizontally oriented.

The distance to these foci from the center is denoted as "c." Using the given foci, we determine that c = √ 10 . This means each focus is √ 10 units away from the center of the hyperbola, (0,0 ). Understanding the foci's placement helps you define the shape and orientation of the hyperbola, as they are fundamental to its structure.

Equation of a Hyperbola

The standard equation for a hyperbola centered at the origin with a horizontal transverse axis is:

• \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

Here, "a" represents the distance from the center to each vertex, while "b" describes the distance from the center along the conjugate axis. The relationship between these parameters and the foci is given by the equation c^2 = a^2 + b^2.
Given our problem parameters, we already know that \( c^2 = 10\), since \(c = \sqrt{10}\). To find the complete equation, we need to compute a^2 and b^2 using the point provided: (4, √18). Substituting these values helps us adjust the equation:

• Place \(x = 4\), \(y = \sqrt{18}\) in the equation and solve for unknowns.

Solving the relationship, using \[b^2 = a^2 + 10\], gives us precise values of a and b to be used in the final hyperbola equation.

Hyperbola Vertices

Vertices of the hyperbola are critical as they represent the closest points of the hyperbola to the center.Using our hyperbola's equation parameters, the vertices are ± a units away from the center along the transverse axis. Since we've determined \(c^2 = a^2 + b^2\), knowing one of these will allow you to calculate the others.
In this exercise, solving c^2 = a^2 + b^2, we eventually derive that:

• a^2 = 4
• b^2 = 6

This makes a, the distance to the vertices (thus the vertices are at (±2, 0)). Identifying the vertices ensures a complete understanding of the hyperbola size and direction along its principal axis. This visualization aids in appreciating the geometric structure of the hyperbola.