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. Here is the step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ ax^2 + 2hxy + by^2 = 0 \]
We differentiate each term with respect to \(x\):
- The derivative of \(ax^2\) is \(2ax\).
- The derivative of \(2hxy\) requires the product rule:
\[ \frac{d}{dx}(2hxy) = 2hy + 2hx\frac{dy}{dx} \]
- The derivative of \(by^2\) is \(2by\frac{dy}{dx}\).

Putting it all together, we have:
\[ 2ax + (2hy + 2hx\frac{dy}{dx}) + (2by\frac{dy}{dx}) = 0 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate the terms involving \(\frac{dy}{dx}\):
\[ 2ax + 2hy + 2hx\frac{dy}{dx} + 2by\frac{dy}{dx} = 0 \]
Combine the \(\frac{dy}{dx}\) terms:
\[ 2ax + 2hy + (2hx + 2by)\frac{dy}{dx} = 0 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we can isolate \(\frac{dy}{dx}\):
\[ (2hx + 2by)\frac{dy}{dx} = - (2ax + 2hy) \]
Dividing both sides by \(2hx + 2by\) gives:
\[ \frac{dy}{dx} = -\frac{2ax + 2hy}{2hx + 2by} \]

### Step 4: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = -\frac{ax + hy}{hx + by} \]

### Step 5: Final result
Thus, the final result for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{ax + hy}{hx + by} \]