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Magazine Chapter 4: Problem 69
Show that the lines \(y=(b / a) x\) and \(y=-(b / a) x\) are slant asymptotes of the hyperbola \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1.\)
Short Answer
Expert verified
The lines are the asymptotes of the hyperbola.
Step by step solution
01
Identify the Hyperbola Form
The given hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This is a standard form of a hyperbola that opens horizontally.
02
Determine Asymptote Equations
A hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) has asymptotes given by the equations \(y = \pm \frac{b}{a} x\). Here, the asymptotes are \(y = \frac{b}{a} x\) and \(y = -\frac{b}{a} x\).
03
Substitute Hyperbola Equations with Asymptotes Form
To show that \(y = \frac{b}{a} x\) and \(-\frac{b}{a} x\) are asymptotes, we simplify the given asymptotic form. Check by substituting \(y = \frac{b}{a} x\) or \(y = -\frac{b}{a} x\) into the hyperbola equation center by applying limits or by simplification.
04
Check Consistency with Hyperbola
For \(y = \frac{b}{a} x\), substitute into the hyperbola equation: \[\frac{x^2}{a^2} - \frac{\left(\frac{b}{a}x\right)^2}{b^2} = 0\]. This simplifies to zero, showing that the slopes conform, confirming \(y = \frac{b}{a}x\) is correct. Similarly, perform for \(y = -\frac{b}{a}x\).
05
Conclude Asymptotes Confirmation
As both lines, \(y = \frac{b}{a} x\) and \(y = -\frac{b}{a} x\), satisfy the asymptotic conditions and simplify to zero, they confirm that they are indeed the slant asymptotes of the given hyperbola.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbola Equations
The concept of a hyperbola can be a bit confusing at first, but let's break it down. A hyperbola is a type of conic section formed by the intersection of a plane and a double-cone.
In simpler terms, imagine slicing a cone at an angle, resulting in a curve that mirrors itself on either side of a central line or axis. The mathematical representation of a hyperbola provides insight into its geometric properties.
In simpler terms, imagine slicing a cone at an angle, resulting in a curve that mirrors itself on either side of a central line or axis. The mathematical representation of a hyperbola provides insight into its geometric properties.
- The equation \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] represents a hyperbola that is oriented horizontally. This means its two 'arms' open left and right.
- The terms \(\frac{x^2}{a^2}\) and \(\frac{y^2}{b^2}\) help determine the size and orientation of these arms.
Slope of Lines
Understanding the slope of a line is essential in geometry, as it describes the line鈥檚 steepness. For a hyperbola's asymptotes, this is particularly useful for determining direction.
For a line described by the equation \(y = mx\), \(m\) represents the slope. In our case, the slopes of the asymptotes for the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are \(\pm \frac{b}{a}\).
For a line described by the equation \(y = mx\), \(m\) represents the slope. In our case, the slopes of the asymptotes for the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are \(\pm \frac{b}{a}\).
- A positive slope, like \(\frac{b}{a}\), indicates that the line rises from left to right.
- A negative slope, such as \(-\frac{b}{a}\), implies the line falls from left to right.
- The magnitude of the slope affects the sharpness of the tilt.
Standard Form of Hyperbola
The standard form of a hyperbola helps quickly identify its key characteristics. The standard form of a hyperbola oriented horizontally is expressed as \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\].
This form is extremely useful in sketching or analyzing the graph of a hyperbola, providing immediate insight into the center, vertices, and asymptotes.
This form is extremely useful in sketching or analyzing the graph of a hyperbola, providing immediate insight into the center, vertices, and asymptotes.
- \(a\) and \(b\) are constants that define the distances from the hyperbola center to the vertices along the x and y axes respectively.
- This form makes it easy to determine the slopes of the asymptotes directly, which are \(\pm \frac{b}{a}\).
- To shift the hyperbola to a different center point, adjust the equation to \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\), with \((h,k)\) being the center.
Slant Asymptotes
Asymptotes are lines that the hyperbola approaches but never actually reaches. For a hyperbola like\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\], the slant asymptotes are objects of keen interest.
Slant asymptotes give a hyperbola its characteristic bow-tie shape by determining the direction of its branches. For these types of hyperbolas, the asymptotes can be directly derived from the equation as \(y = \pm \frac{b}{a} x\).
Slant asymptotes give a hyperbola its characteristic bow-tie shape by determining the direction of its branches. For these types of hyperbolas, the asymptotes can be directly derived from the equation as \(y = \pm \frac{b}{a} x\).
- The slopes \(\pm \frac{b}{a}\) determine the angle at which the asymptotes intersect the axes.
- These lines act as boundaries that the arms of the hyperbola continually approach, but never quite touch.
