Question:
The asymptotes parallel to \(x\)-axis of the curve \(y^3+x^2y+2xy^2-y+1=0\) is:
The asymptotes parallel to \(x\)-axis of the curve \(y^3+x^2y+2xy^2-y+1=0\) is:
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For asymptotes parallel to \(x\)-axis, put the coefficient of highest power of \(x\) equal to zero.
Updated On: May 19, 2026
- \(y=1\)
- \(y=2\)
- \(y=0\)
- \(y=-1\)
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The Correct Option is C
Solution and Explanation
Concept:
An asymptote parallel to \(x\)-axis has the form: \[ y=c \]
Step 1: Given curve. \[ y^3+x^2y+2xy^2-y+1=0 \]
Step 2: For horizontal asymptote.
For an asymptote parallel to \(x\)-axis, take \(x\to\infty\) and \(y\) finite. The highest power term in \(x\) is: \[ x^2y \] For the expression to remain finite near an asymptote, the coefficient of \(x^2\) must vanish. \[ y=0 \] Therefore, the horizontal asymptote is: \[ y=0 \] \[ \therefore \text{Correct Answer is (C)} \]
An asymptote parallel to \(x\)-axis has the form: \[ y=c \]
Step 1: Given curve. \[ y^3+x^2y+2xy^2-y+1=0 \]
Step 2: For horizontal asymptote.
For an asymptote parallel to \(x\)-axis, take \(x\to\infty\) and \(y\) finite. The highest power term in \(x\) is: \[ x^2y \] For the expression to remain finite near an asymptote, the coefficient of \(x^2\) must vanish. \[ y=0 \] Therefore, the horizontal asymptote is: \[ y=0 \] \[ \therefore \text{Correct Answer is (C)} \]
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