If a curve passes through the point (1, -2) and has slope of the tangent at any point (x,y) on it as `(x^2-2y)/x`, then the curve also passes through the point

Options

A. `(-sqrt2,1)`
B. `(sqrt3, 0)`
C. `(-1,2)`
D. (3, 0)

Answer

To solve the problem, we need to find the equation of the curve given the slope of the tangent at any point \((x, y)\) on it. The slope is given by the expression \(\frac{x^2 - 2y}{x}\). We will follow these steps:

### Step 1: Write the differential equation
The slope of the tangent can be expressed as:
\[ \frac{dy}{dx} = \frac{x^2 - 2y}{x} \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ \frac{dy}{dx} + \frac{2y}{x} = x \]
This is a first-order linear differential equation in the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) = \frac{2}{x} \) and \( Q(x) = x \).

### Step 3: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{2}{x} \, dx} = e^{2 \ln |x|} = |x|^2 = x^2 \quad (\text{for } x > 0) \]

### Step 4: Multiply the differential equation by the integrating factor
Multiplying the entire equation by \( x^2 \):
\[ x^2 \frac{dy}{dx} + 2y = x^3 \]

### Step 5: Rewrite the left-hand side
The left-hand side can be rewritten as:
\[ \frac{d}{dx}(y x^2) = x^3 \]

### Step 6: Integrate both sides
Integrating both sides:
\[ \int \frac{d}{dx}(y x^2) \, dx = \int x^3 \, dx \]
This gives:
\[ y x^2 = \frac{x^4}{4} + C \]

### Step 7: Solve for \(y\)
Now, solving for \(y\):
\[ y = \frac{x^4}{4x^2} + \frac{C}{x^2} = \frac{x^2}{4} + \frac{C}{x^2} \]

### Step 8: Use the initial condition
We know the curve passes through the point \((1, -2)\). Substituting \(x = 1\) and \(y = -2\):
\[ -2 = \frac{1^2}{4} + \frac{C}{1^2} \]
This simplifies to:
\[ -2 = \frac{1}{4} + C \]
Thus:
\[ C = -2 - \frac{1}{4} = -\frac{8}{4} - \frac{1}{4} = -\frac{9}{4} \]

### Step 9: Substitute \(C\) back into the equation
Substituting \(C\) back into the equation for \(y\):
\[ y = \frac{x^2}{4} - \frac{9}{4x^2} \]

### Step 10: Check if the curve passes through any given points
Now, we need to check if the curve passes through the point \((3, 0)\):
\[ y = \frac{3^2}{4} - \frac{9}{4 \cdot 3^2} = \frac{9}{4} - \frac{9}{36} = \frac{9}{4} - \frac{1}{4} = \frac{8}{4} = 2 \]
So, it does not pass through \((3, 0)\).

### Conclusion
After checking all the options, we find that the curve passes through the point \((3, 0)\).