Let a smooth curve y = f(x) be such that the slope of the tangent at any point (x, y) on it is direc

Q: Wait, why do we use $\ln C$ instead of just $C$ as the constant of integration?

That is a fantastic question! When you integrate both sides of the equation, you get $\ln|y| = -k \ln|x| + C'$. While $C'$ is a perfectly valid constant, using $\ln C$ instead makes the algebra much cleaner. Because $C$ is just a constant, $\ln C$ is also a constant. This allows us to use the logarithmic property $\ln a + \ln b = \ln(ab)$ to combine the terms, which makes it much easier to exponentiate both sides and isolate $y$.

Q: Why can't we just use energy conservation or some other physics principle here?

I love that you are looking for physical connections! While this problem is framed as a curve, it is purely a mathematical exercise in differential equations. The 'physics' here is the geometric definition of the curve's slope. In JEE Advanced, you will often encounter problems that are mathematically identical to physical systems, but here, the 'law' is simply the given relationship between the slope and the coordinates. Trust the math!

Q: How do I know if $k$ should be positive or negative?

Great observation! The sign of $k$ is determined by the data points provided. When you substitute the points $(1, 2)$ and $(8, 1)$ into your derived equation $y = C x^{-k}$, the algebra will naturally dictate the value of $k$. If you follow the steps carefully, you will find that $k = 1/3$. There is no need to guess the sign beforehand; let the points guide you to the correct value.

Question

Let a smooth curve $y = f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\frac{-y}{x}$. If the curve passes through the point $(1, 2)$ and $(8, 1)$, then $|y(\frac{1}{8})|$ is equal to

Options:

(A) $2\log_e 2$

(B) 4

(C) 1

(D) $4\log_e 2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a blank coordinate plane. A smooth, mysterious curve, $y = f(x)$, is waiting to be discovered. We have two vital clues: the curve passes through the points $(1, 2)$ and $(8, 1)$.

Our mission is to find the value of $|y(1/8)|$. This is a detective story where we use the language of calculus to uncover the identity of this curve.

### The Differential Equation: The Heart of the Curve

The problem states that the slope of the tangent at any point $(x, y)$ is directly proportional to $-y/x$. In the language of calculus, the slope is the derivative $dy/dx$.

We can express this relationship as:

$$\frac{dy}{dx} = k \left( -\frac{y}{x} ight)$$

Here, $k$ is our constant of proportionality. This differential equation acts as the DNA of our curve, dictating how it bends and moves at every point.

### The Art of Separation

To solve this, we use the technique of separation of variables. We isolate the $y$ terms on one side and the $x$ terms on the other:

$$\frac{dy}{y} = -k \frac{dx}{x}$$

Now, we apply the power of integration to both sides:

$$\int \frac{1}{y} \, dy = -k \int \frac{1}{x} \, dx$$

Integrating both sides yields $\ln|y| = -k \ln|x| + \ln C$, where $\ln C$ is our constant of integration. This choice of constant simplifies the algebraic manipulation significantly.

### Unveiling the Equation

Using the properties of logarithms, we rewrite the right side as $\ln|C x^{-k}|$. By exponentiating both sides, we strip away the logarithms to reveal the general power function:

$$y = C x^{-k}$$

We now determine the constants using our anchor points. Substituting $(1, 2)$ into the equation:

$$2 = C(1)^{-k} \implies C = 2$$

Next, we use the point $(8, 1)$ to find $k$:

$$1 = 2(8)^{-k} \implies \frac{1}{2} = 8^{-k} \implies 2 = 8^k$$

Since $8 = 2^3$, we have $2^1 = (2^3)^k$, which implies $1 = 3k$. Thus, $k = 1/3$.

### Final Calculation

The specific equation of our curve is:

$$y = 2 x^{-1/3}$$

To find $|y(1/8)|$, we substitute $x = 1/8$ into our equation:

$$y\left(\frac{1}{8}ight) = 2 \left( \frac{1}{8} ight)^{-1/3}$$

Applying the rules of exponents, this simplifies to:

$$y\left(\frac{1}{8}ight) = 2(8)^{1/3} = 2 	imes 2 = 4$$

The absolute value of $4$ is simply **4**. We have successfully navigated the differential equation to arrive at the final result.

Select Answer:

Visualized Solution

Visualizing the Problem

Defining the Slope Relationship

Separation of Variables

Integrating Both Sides

Simplifying the Logarithms

Finding Constant C

Finding Constant k

Solving for k using Powers

The Final Equation

Evaluating at x = \frac{1}{8}

Final Calculation

The Sigma Insight: Variable Separable Method

Analyzing the Setup

The Differential Equation

The Art of Separation

Unveiling the Equation

Final Calculation

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Select Answer:

Visualized Solution

Visualizing the Problem

Defining the Slope Relationship

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