A curve is such that the ratio of the subnomal at any point to the sum of its co-ordinates is equal tothe ratio of the ordinate of this point to its abscissa. If the curve passes through M(1,0), then possible equation of the curve is(are)

Options

A. `y= x ln x`
B. `y=(ln x)/(x)`
C. `y= (2 (x-1))/(x ^(2))`
D. `y= (1-x ^(2))/(2x)`

Answer

To solve the problem, we need to derive the equation of the curve based on the given conditions. Let's break it down step by step.

### Step 1: Understand the relationship given in the problem

The problem states that the ratio of the subnormal at any point to the sum of its coordinates is equal to the ratio of the ordinate of this point to its abscissa.

Let the coordinates of the point on the curve be \( (x, y) \).

### Step 2: Define the subnormal

The subnormal \( N \) at a point on the curve is given by:
\[ N = y \cdot \frac{dy}{dx} \]

### Step 3: Set up the equation based on the given ratio

According to the problem, we have:
\[ \frac{N}{x + y} = \frac{y}{x} \]

Substituting the expression for \( N \):
\[ \frac{y \cdot \frac{dy}{dx}}{x + y} = \frac{y}{x} \]

### Step 4: Cross-multiply to eliminate the fraction

Cross-multiplying gives us:
\[ y \cdot \frac{dy}{dx} \cdot x = y \cdot (x + y) \]

Assuming \( y \neq 0 \), we can divide both sides by \( y \):
\[ x \cdot \frac{dy}{dx} = x + y \]

### Step 5: Rearranging the equation

Rearranging the equation, we have:
\[ x \cdot \frac{dy}{dx} - y = x \]

### Step 6: Separate the variables

We can rewrite this as:
\[ \frac{dy}{dx} = \frac{y + x}{x} \]

### Step 7: Substitute \( y = xt \)

Let \( y = xt \), where \( t \) is a function of \( x \). Then:
\[ \frac{dy}{dx} = t + x \frac{dt}{dx} \]

Substituting into the equation gives:
\[ t + x \frac{dt}{dx} = \frac{xt + x}{x} \]

This simplifies to:
\[ t + x \frac{dt}{dx} = t + 1 \]

### Step 8: Isolate the derivative

Subtract \( t \) from both sides:
\[ x \frac{dt}{dx} = 1 \]

### Step 9: Separate variables again

Rearranging gives:
\[ \frac{dt}{dx} = \frac{1}{x} \]

### Step 10: Integrate both sides

Integrating both sides:
\[ \int dt = \int \frac{1}{x} dx \]
\[ t = \ln |x| + C \]

### Step 11: Substitute back for \( y \)

Recall that \( t = \frac{y}{x} \), so:
\[ \frac{y}{x} = \ln |x| + C \]
\[ y = x(\ln |x| + C) \]

### Step 12: Use the point M(1, 0) to find C

Since the curve passes through the point \( M(1, 0) \):
\[ 0 = 1(\ln(1) + C) \]
\[ 0 = 0 + C \implies C = 0 \]

### Final Equation of the Curve

Thus, the equation of the curve is:
\[ y = x \ln |x| \]