- 2073 Chaitra
Define and classify asymptotes of a curve
Short answer
Types include horizontal (y = k), vertical (x = h), and oblique (y = mx + c, m ≠ 0). Also by inspection and for polar curves where r → ∞.
State the types of asymptotes to a curve.
An asymptote of a curve is a straight line such that the distance of a point on the curve from the line tends to zero as the point goes to infinity along the curve. It is the tangent to the curve at infinity.
Types
- Asymptote parallel to the x-axis (horizontal): a line y = k. It is found by equating to zero the coefficient of the highest power of x in the equation of the curve.
- Asymptote parallel to the y-axis (vertical): a line x = h. It is found by equating to zero the coefficient of the highest power of y.
- Oblique asymptote (inclined): a line y = mx + c with m ≠ 0, ∞. The slopes m are the roots of φₙ(m) = 0, where φₙ is the highest-degree part of the equation, and c follows from φₙ₋₁(m) + c φₙ′(m) = 0.
Besides these, a curve may have asymptotes by inspection (for example a hyperbola's asymptotes differ from the curve only by a constant) and, in advanced work, circular or curvilinear asymptotes. For polar curves, an asymptote is found from r → ∞ at the angle where 1/f(θ) = 0.
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