Let M and N be the number of points on the curve y⁵-9 x y+2 x=0, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M+N equals ___________.

Solution

Maths sir · 5 steps

Set the numerator or denominator of dy/dx to zero, then check the point really lies on the curve.

1. Find the slope. Differentiate y⁵ - 9xy + 2x = 0 implicitly: 5y⁴ y' - 9y - 9xy' + 2 = 0. This gives the slope below.

   dy/dx = (9y - 2)/(5y⁴ - 9x)

2. Parallel to x-axis. The numerator must be zero, so y = 2/9. Put this in the curve: (2/9)⁵ - 2x + 2x = (2/9)⁵ ≠ 0. No point on the curve has y = 2/9, so M = 0.
3. Parallel to y-axis. The denominator must be zero, so 9x = 5y⁴. Put this in the curve: y⁵ - 5y⁵ + 10/9y⁴ = 0, which gives y⁴(10/9 - 4y) = 0, so y = 0 or y = 5/18.
4. Check both points. At (0, 0) the numerator is -2 ≠ 0, and at y = 5/18 it is 1/2 ≠ 0. So both are true vertical tangents, and N = 2.
5. So. M + N = 0 + 2 = 2.

The common slip: 3. Taking y = 2/9 as a horizontal tangent without checking that it lies on the curve gives M = 1 and a total of 3.

The idea to keep. Solving slope = 0 or slope undefined only gives a condition. A tangent exists only where that condition meets the curve itself, so always substitute back.

Answer from the NTA final answer key. Solution by PrepWiser.