The equation of curves which intersect the hyperbola `xy=4` at an angle `pi/2` is

Answer

To find the equation of curves that intersect the hyperbola \(xy = 4\) at an angle of \(\frac{\pi}{2}\) (90 degrees), we can follow these steps:

### Step 1: Understand the condition of intersection
The curves intersect at an angle of \(\frac{\pi}{2}\), which means the slopes of the tangents to the curves at the point of intersection are negative reciprocals of each other. If \(f(x)\) is one curve and \(g(x)\) is the other, then:
\[ f'(x) \cdot g'(x) = -1 \]

### Step 2: Express the hyperbola in terms of \(y\)
The equation of the hyperbola is given by:
\[ xy = 4 \]
We can express \(y\) in terms of \(x\):
\[ y = \frac{4}{x} \]

### Step 3: Find the derivative of the hyperbola
To find the slope of the tangent to the hyperbola, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = -\frac{4}{x^2} \]
Thus, the slope of the tangent to the hyperbola at any point is:
\[ f'(x) = -\frac{4}{x^2} \]

### Step 4: Set up the equation for the other curve
Let the slope of the other curve \(g(x)\) be \(g'(x)\). From the perpendicularity condition, we have:
\[ -\frac{4}{x^2} \cdot g'(x) = -1 \]
This simplifies to:
\[ g'(x) = \frac{x^2}{4} \]

### Step 5: Integrate to find \(g(x)\)
Now, we integrate \(g'(x)\) to find \(g(x)\):
\[ g(x) = \int \frac{x^2}{4} \, dx = \frac{x^3}{12} + C \]
where \(C\) is the constant of integration.

### Step 6: Write the final equation of the curves
Thus, the required curves that intersect the hyperbola \(xy = 4\) at an angle of \(\frac{\pi}{2}\) are given by:
\[ g(x) = \frac{x^3}{12} + C \]

### Final Answer
The equation of the curves is:
\[ y = \frac{x^3}{12} + C \]

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