Find the condition that curves `2x=y^(2) and 2xy=k` intersect orthogonally.

Answer

To find the condition that the curves \(2x = y^2\) and \(2xy = k\) intersect orthogonally, we will follow these steps:

### Step 1: Differentiate the first curve
The first curve is given by the equation:
\[ 2x = y^2 \]
Differentiating both sides with respect to \(x\):
\[ 2 \frac{dx}{dx} = 2y \frac{dy}{dx} \]
This simplifies to:
\[ 2 = 2y \frac{dy}{dx} \]
Thus, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{1}{y} \]

### Step 2: Find the slope at the intersection point
Let the point of intersection be \((x_1, y_1)\). The slope of the tangent to the first curve at this point is:
\[ m_1 = \frac{1}{y_1} \]

### Step 3: Differentiate the second curve
The second curve is given by:
\[ 2xy = k \]
Differentiating both sides with respect to \(x\):
\[ 2y + 2x \frac{dy}{dx} = 0 \]
Rearranging gives:
\[ 2x \frac{dy}{dx} = -2y \]
Thus, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 4: Find the slope at the intersection point for the second curve
At the point of intersection \((x_1, y_1)\), the slope of the tangent to the second curve is:
\[ m_2 = -\frac{y_1}{x_1} \]

### Step 5: Set the condition for orthogonality
The curves intersect orthogonally if the product of their slopes is \(-1\):
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \(m_1\) and \(m_2\):
\[ \left(\frac{1}{y_1}\right) \left(-\frac{y_1}{x_1}\right) = -1 \]
This simplifies to:
\[ -\frac{1}{x_1} = -1 \]
Thus, we have:
\[ x_1 = 1 \]

### Step 6: Find \(y_1\) using the first curve
Substituting \(x_1 = 1\) into the first curve equation:
\[ 2(1) = y_1^2 \]
This simplifies to:
\[ y_1^2 = 2 \quad \Rightarrow \quad y_1 = \pm \sqrt{2} \]

### Step 7: Substitute into the second curve to find \(k\)
Now, we substitute \(x_1 = 1\) and \(y_1 = \sqrt{2}\) into the second curve equation:
\[ 2xy = k \quad \Rightarrow \quad 2(1)(\sqrt{2}) = k \]
Thus:
\[ k = 2\sqrt{2} \]

### Final Condition
The condition for the curves \(2x = y^2\) and \(2xy = k\) to intersect orthogonally is:
\[ k = 2\sqrt{2} \]