Let $$M$$ and $$N$$ be the number of points on the curve $$y^5 - 9xy + 2x = 0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M + N$$ equals

We need to find the distance of the point $$(7, 1)$$ from the line $$6x + y - 46 = 0$$.

The formula for the distance from a point $$(x_0, y_0)$$ to a line $$ax + by + c = 0$$ is $$d = \dfrac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$$.

Here $$a = 6$$, $$b = 1$$, $$c = -46$$, and $$(x_0, y_0) = (7, 1)$$.

$$d = \dfrac{|6(7) + 1(1) - 46|}{\sqrt{36 + 1}} = \dfrac{|42 + 1 - 46|}{\sqrt{37}} = \dfrac{|-3|}{\sqrt{37}} = \dfrac{3}{\sqrt{37}}$$

Since the distance is given as $$\dfrac{a}{\sqrt{37}}$$, comparing we get $$a = 3$$.

The answer is $$3$$.