About Asymptote Formula
A straight line w.r.t a curve that tends to meet the curve at infinity is called an asymptote. A line drawn at a minimum parallel distance to the tangent of a curve that does not cut or touch the curve can be more readily comprehended. For a hyperbola, the asymptote formula is commonly defined. The Asymptote Formula is written as a line equation.
What is an Asymptote Formula?
A hyperbola's asymptote is a pair of straight lines. A hyperbola's asymptotes using an equation x2/a2 - y2/b2 = 0 is given by the following formula:
Asymptotes equation: y = b/a.x, and y = -b/a.x
Pair of Asymptotes equation: x2/a2 - y2/b2 = 0
Exercising the Asymptote Formula
Example 1: : Find the equation for a pair of hyperbola asymptotes. x2/16 - y2/25 = 1.
Solution:
Hyperbola given equation is x2/16 - y2/25 = 1
An equation is needed for a hyperbola x2/a2 - y2/b2 = 1
Its pair of asymptotes equation is bx/a, -bx/a
Here we know that a = 4 and b = 5
Thus, equation of pair of asymptotes are y = 4x/5 and y =-4x/5
Example 2: Find the equations of asymptotes of hyperbola given as: x2/49 - y2/36 = 1.
Solution:
The given equation of the hyperbola is x2/49 - y2/36 = 1
this emply: x2/72 - y2/62 = 1
Now compare the above equation with the standard equation of a hyperbola x2/a2 - y2/b2 = 1
We have a = 7 and b = 6
Further the equations of the asymptotes is y = b/a.x, and y = -b/a.x
y = 6x/7 and y = -6x/7
7y = 6x and 7y = -6x
7y- 6x = 0 and 6x + 7y = 0
Thus, the equations of the asymptotes are 7y- 6x = 0 and 6x + 7y = 0
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Frequently Asked Questions
An asymptote is a line that a curve approaches but never touches. The formula for asymptotes depends on the type:
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Vertical Asymptote: Found by setting the denominator of a rational function to zero, x=ax = ax=a.
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Horizontal Asymptote: Determined by comparing the degrees of the numerator and denominator in a rational function.
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Oblique Asymptote: Found using polynomial long division when the numerator's degree is one higher than the denominator.
- Vertical Asymptotes occur where the function is undefined, typically at values making the denominator zero. Example: For f(x)=1x−3f(x) the vertical asymptote is x=3x = 3x=3.
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Horizontal Asymptotes are found by analyzing the highest degree terms in a rational function. Example: For f(x)=2xx+1f(x) as x→∞ the horizontal asymptote is y=2y
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