If `ax^(2)+2hxy+by^(2)=0` then `(dy)/(dx)` is

Options

A. `(y)/(x)`
B. `(x)/(y)`
C. `-(x)/(x)`
D. None of these

Answer

To find \(\frac{dy}{dx}\) for the equation \(ax^2 + 2hxy + by^2 = 0\), we will use implicit differentiation. Let's go through the solution step by step.

### Step 1: Differentiate the equation implicitly
We start with the equation:
\[ ax^2 + 2hxy + by^2 = 0 \]
Now, we differentiate each term with respect to \(x\).

### Step 2: Differentiate each term
1. The derivative of \(ax^2\) with respect to \(x\) is:
\[ \frac{d}{dx}(ax^2) = 2ax \]

2. The derivative of \(2hxy\) with respect to \(x\) requires the product rule:
\[ \frac{d}{dx}(2hxy) = 2h\left(x\frac{dy}{dx} + y\right) \]
Here, we differentiate \(x\) and multiply by \(y\), and differentiate \(y\) and multiply by \(x\).

3. The derivative of \(by^2\) with respect to \(x\) is:
\[ \frac{d}{dx}(by^2) = 2by\frac{dy}{dx} \]

### Step 3: Combine the derivatives
Putting it all together, we have:
\[ 2ax + 2h\left(x\frac{dy}{dx} + y\right) + 2by\frac{dy}{dx} = 0 \]

### Step 4: Simplify the equation
Expanding this gives:
\[ 2ax + 2hx\frac{dy}{dx} + 2hy + 2by\frac{dy}{dx} = 0 \]
Now, we can group the terms involving \(\frac{dy}{dx}\):
\[ 2hx\frac{dy}{dx} + 2by\frac{dy}{dx} = -2ax - 2hy \]

### Step 5: Factor out \(\frac{dy}{dx}\)
Factoring out \(\frac{dy}{dx}\) gives:
\[ \frac{dy}{dx}(2hx + 2by) = -2ax - 2hy \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{-2ax - 2hy}{2hx + 2by} \]
Simplifying this, we get:
\[ \frac{dy}{dx} = \frac{-(ax + hy)}{hx + by} \]

### Step 7: Rewrite the expression
We can rewrite this as:
\[ \frac{dy}{dx} = -\frac{ax + hy}{hx + by} \]

### Final Result
Thus, the final expression for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{ax + hy}{hx + by} \]