Let M and N be the number of points on the curve y⁵ - 9xy + 2x = 0, where the tangents to the curve

Q: Wait, why do we reject the horizontal tangent case? I found $y = \frac{2}{9}$, so shouldn't that be a point?

That is a classic trap! Finding a $y$-value that makes the slope zero is only the first half of the battle. You must ensure that this $y$-value is actually part of the curve. When you substitute $y = \frac{2}{9}$ into the original equation, the $x$ terms cancel out, leaving you with a contradiction: $(\frac{2}{9})^5 = 0$. This proves that no point on the curve exists with that $y$-coordinate. Always verify your points!

Q: Why can't we just use energy conservation or some other shortcut?

This is a pure calculus problem about the geometry of a curve. Energy conservation applies to physical systems, but here we are dealing with the algebraic properties of a function defined implicitly. The only way to find tangents is through the derivative, which tells us the instantaneous slope at any point.

Q: How do I know if the denominator is zero at the same time as the numerator?

Great question. If both the numerator and denominator are zero at the same point, the slope is indeterminate ($0/0$), and you would need to use limits to find the actual slope. In this problem, we checked the numerator at our vertical tangent points ($y=0$ and $y=5/18$) and found it was non-zero, so we are safe!

Question

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 ________

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the mathematical landscape. Today, we are going to explore the curve defined by the equation $y^5 - 9xy + 2x = 0$.

This is not just an equation; it is a map of a hidden path in the Cartesian plane. Our mission is to find $M$, the number of points where the tangent is horizontal, and $N$, the number of points where the tangent is vertical.

Finally, we will sum them up to find our destination.

### The Toolkit: Implicit Differentiation

To understand the tangents, we need the slope, $\frac{dy}{dx}$. Because our curve is defined implicitly, we cannot easily isolate $y$.

Instead, we use implicit differentiation. Differentiating $y^5 - 9xy + 2x = 0$ with respect to $x$, we get:

$$5y^4 \frac{dy}{dx} - 9\left(y + x \frac{dy}{dx}ight) + 2 = 0$$

By grouping the $\frac{dy}{dx}$ terms, we arrive at the master slope equation:

$$\frac{dy}{dx} = \frac{9y - 2}{5y^4 - 9x}$$

This fraction is our compass.

### The Horizontal Hunt ($M$)

A horizontal tangent means the slope is zero. For a fraction to be zero, the numerator must be zero.

Thus, we set $9y - 2 = 0$, which gives us $y = \frac{2}{9}$. But hold on! We must check if this point exists on the curve.

Substituting $y = \frac{2}{9}$ into the original equation $y^5 - 9xy + 2x = 0$, we get:

$$\left(\frac{2}{9}ight)^5 - 9x\left(\frac{2}{9}ight) + 2x = 0$$

The $x$ terms simplify to $-2x + 2x$, which cancel out completely. We are left with $(\frac{2}{9})^5 = 0$, which is impossible.

Therefore, there are no points with horizontal tangents. Thus, **$M = 0$**.

### The Vertical Quest ($N$)

A vertical tangent occurs when the slope is undefined, meaning the denominator of our derivative must be zero. We set $5y^4 - 9x = 0$, which implies:

$$x = \frac{5y^4}{9}$$

Now, we substitute this back into the original curve:

$$y^5 - 9\left(\frac{5y^4}{9}ight)y + 2\left(\frac{5y^4}{9}ight) = 0$$

Simplifying this, we get $y^5 - 5y^5 + \frac{10y^4}{9} = 0$, or $-4y^5 + \frac{10y^4}{9} = 0$. Factoring out $y^4$, we find:

$$y^4\left(-4y + \frac{10}{9}ight) = 0$$

This gives us two solutions: $y = 0$ and $y = \frac{5}{18}$.

We verify these points: at $y=0$, $x=0$, and the numerator $9(0)-2 = -2 
eq 0$. At $y=\frac{5}{18}$, $x$ is a valid real number, and the numerator $9(\frac{5}{18}) - 2 = \frac{1}{2} 
eq 0$.

Both points are valid! Thus, **$N = 2$**.

### The Final Victory

We have navigated the traps and discovered the truth. With $M = 0$ and $N = 2$, the sum $M + N$ is exactly **$2$**.

You have successfully mapped the tangents of this curve. Keep this curiosity alive, for it is the fuel of every great mathematician.

Animated Solution for Mathematics - Differentiation: Let $M$ and $N$ be the number of points on the curve $y^{5} - 9 x y + 2 x = 0$ , where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of $M + N$ equals ________

Enter Numerical Value:

Visualized Solution

The Curve $y^{5} - 9 x y + 2 x = 0$

Implicit Differentiation

Finding $\frac{d y}{d x}$

Horizontal Tangents ( $M$ )

Testing $y = \frac{2}{9}$

Conclusion for $M$

Vertical Tangents ( $N$ )

Substituting $x = \frac{5 y ^{4}}{9}$

Solving for $y$

Verifying Points

Final Value of $M + N$

The Sigma Insight: Tangents, Normals and Rate Measure

Analyzing the Setup

The Toolkit

The Horizontal Hunt ( $M$ )

The Vertical Quest ( $N$ )

The Final Victory

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Enter Numerical Value:

Visualized Solution

The Curve .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y5−9xy+2x=0

Implicit Differentiation

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } dxdy​

Horizontal Tangents ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M)

Testing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y=92​

Conclusion for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M

Vertical Tangents ( .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } N)

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=95y4​

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y