Problem 46 - Precalculus: Mathematics for Calculus

Question

Find an equation for the hyperbola that satisfies the given conditions. Asymptotes: \(y=\pm x,\) hyperbola passes through \((1,2)\)

Step-by-Step Solution

Answer

The problem cannot produce valid \(a^2\) in standard asymptote conditions; rethink or reevaluate calculations.

Step 1: Understand the Asymptotes

The given asymptotes are lines: \(y = x\) and \(y = -x\). This suggests that the hyperbola is centered at the origin \((0,0)\) and is a standard hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0\). Since the slopes of the asymptotes are \(\pm 1\), it indicates that \(a = b\).

Step 2: Formulate the Hyperbola Equation

Given that \(a = b\), we know the standard form of the equation is \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\), which simplifies to \(\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\), or \(x^2 - y^2 = a^2\cdot 1\).

Step 3: Use the Given Point

Substitute the point \((1, 2)\), which lies on the hyperbola, into the equation \(x^2 - y^2 = a^2\): \(1^2 - 2^2 = a^2\). Compute to get \(1 - 4 = a^2\), which gives \(a^2 = -3\). This is not valid for a hyperbola since \(a^2\) must be positive.

Step 4: Correct the Mistake in Formulation

Upon realizing the error from \(a^2 = -3\), re-evaluate the symmetric property of hyperbola: Use \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) with permissible values for \(a\) and \(b\) respecting the asymptotes. Adjust the attempt and revise errors: reconsider possible calculation or question representation to omit an erroneous situation from a basic symmetrical check.

Step 5: Verify or Provide Task Clarification

Re-analyzing confirms the impossibility given usual calculations in this form unless specific value inquiry resolvable external to calculation vicinity. Thus, when derived upon logical conclusion and calculation impossibility, adjust the verification either: on starting early with coordinate check consistency, re-reading conditions or allowing adjustment options through reevaluation for the apparent solution pathway.

Key Concepts

Asymptotes

Understanding asymptotes is essential when studying hyperbolas. Asymptotes are imaginary lines that a hyperbola approaches but never touches. In the context of hyperbolas, these lines give us valuable information about the orientation and the shape of a hyperbola.

• The asymptotes for the given hyperbola are expressed with equations: \(y = x\) and \(y = -x\).
• This suggests that the hyperbola is symmetrically centered around the origin.
• The fact that both asymptotes have a slope of \(\pm 1\) indicates equal values for the semi-major and semi-minor axes, meaning \(a = b\).

Since these asymptotes intersect at the origin, the center of this hyperbola is at \((0,0)\), helping us set up the basic framework for our hyperbola equation.

Standard Form of Hyperbola

The standard form of a hyperbola is crucial to derive, analyze, and manipulate its equations depending on the orientation and the axis.For hyperbolas, we generally work with one of two standard forms:

• If the hyperbola opens along the x-axis: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• If the hyperbola opens along the y-axis: \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

In our example, because of the asymptotes \(y = \pm x\) and given \(a = b\), the appropriate standard form here is \( \frac{x^2}{a^2} - \frac{y^2}{a^2} = 1 \) which simplifies to \(x^2 - y^2 = a^2\). This form tells us the hyperbola opens along the x-axis, confirming its orientation and aiding in further calculations.

Hyperbola Center

The center of a hyperbola is a critical point that informs the symmetry of the shape. For many hyperbolas, the center can be deduced from the intersection of the asymptotes.
In this particular problem where the asymptotes are \(y = x\) and \(y = -x\), the center of the hyperbola naturally occurs at

• The point (0,0). This tells us that the hyperbola is symmetric about both axes in the coordinate plane, making this a key point in our equation.
• The center acts as a reference from which the rest of the hyperbola stretches out in a rectangular outline defined by the asymptotes.

This symmetrical property is used again when verifying solutions and checking consistency in calculations.

Solving Hyperbola Equations

Solving the equations of a hyperbola involves substituting known values and ensuring all theoretical properties hold true. In our task, the hyperbola equation is a bit tricky due to the initial wrong formulation.To solve, follow these steps:

• Use the derived equation \(x^2 - y^2 = a^2\) and substitute the given point \((1,2)\).
• Compute: \(1^2 - 2^2 = a^2\), leading to \(1 - 4 = a^2\).
• End up with \(a^2 = -3\), an impossibility since \(a^2\) must be positive in real solutions.

Upon realizing an unreasonable result, a reevaluation of calculations or assumptions must occur. Errors might stem from constraints not aligning or assumptions like equal axes needing reconsideration when computing mathematically viable results. Carefully reviewing conditions or restructuring approach might align closer to permitted geometric and algebraic norms in hyperbolas.