The point(s) on the curve , where the tangent is vertical, is (are)
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
Differentiating implicitly: 3y²(dy/dx) + 6x = 12(dy/dx), so dy/dx = 6x/(12 - 3y²) = 2x/(4 - y²).
The tangent is vertical when dx/dy = 0, i.e., the denominator of dy/dx = 0, which means 4 - y² = 0, so y = ±2.
For y = 2: substituting into the curve gives 8 + 3x² = 24, so 3x² = 16, x = ±4/√3. These points (±4/√3, 2) lie on the curve.
For y = -2: substituting gives -8 + 3x² = -24, so 3x² = -16, which has no real solution. Thus the answer is (±4/√3, 2).
The tangent is vertical when dx/dy = 0, i.e., the denominator of dy/dx = 0, which means 4 - y² = 0, so y = ±2.
For y = 2: substituting into the curve gives 8 + 3x² = 24, so 3x² = 16, x = ±4/√3. These points (±4/√3, 2) lie on the curve.
For y = -2: substituting gives -8 + 3x² = -24, so 3x² = -16, which has no real solution. Thus the answer is (±4/√3, 2).
[AIEEE 2002]