Question

Let M and N be the number of points on the curve y 5 - 9 xy + 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 ________.

Answer 1

The correct answer is 2
Here equation of curve is
y 5 – 9 xy + 2 x = 0 …(i)
On differentiating:
$5y^4 \frac{dy}{dx} - 9y - 9x \frac{dy}{dx} + 2 = 0$
∴ $\frac{dy}{dx} = \frac{9y - 2}{5y^4 - 9x}$
When tangents are parallel to x axis then 9 y – 2 = 0
∴ M = 1.
For tangent perpendicular to x-axis
5 y 4 – 9 x = 0 …(ii)
From equation (1) and equation (2) we get only one point.
∴ N = 1.
∴ M + N = 2.