If` (a/b)^(1/3) + (b/a)^(1/3) = sqrt3/2`, then the angle of intersection of the parabola `y^2 = 4ax and x^2 = 4by` at the point other than the origin is

Options

A. `pi/4`
B. `pi/2`
C. `pi/3`
D. none of these

Explanation

To solve the problem, we need to find the angle of intersection of the parabolas \( y^2 = 4ax \) and \( x^2 = 4by \) at the point other than the origin, given that \( \left( \frac{a}{b} \right)^{1/3} + \left( \frac{b}{a} \right)^{1/3} = \frac{\sqrt{3}}{2} \).

### Step-by-Step Solution:

**Step 1: Identify the equations of the parabolas.**
The equations given are:
1. \( y^2 = 4ax \) (Parabola opening to the right)
2. \( x^2 = 4by \) (Parabola opening upwards)

**Hint:** Recognize the standard forms of the parabolas to understand their orientations.

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**Step 2: Find the slopes of the tangents to the parabolas at the intersection point.**
To find the slopes of the tangents at the intersection point, we differentiate both equations.

For \( y^2 = 4ax \):
- Differentiate implicitly: \( 2y \frac{dy}{dx} = 4a \)
- Thus, \( \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \)

For \( x^2 = 4by \):
- Differentiate implicitly: \( 2x = 4b \frac{dy}{dx} \)
- Thus, \( \frac{dy}{dx} = \frac{2x}{4b} = \frac{x}{2b} \)

**Hint:** Use implicit differentiation to find the slopes of the tangents.

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**Step 3: Find the coordinates of the intersection point.**
To find the intersection points, we can set \( y^2 = 4ax \) equal to \( x^2 = 4by \). From the first equation, we can express \( x \) in terms of \( y \):
- From \( y^2 = 4ax \), we have \( x = \frac{y^2}{4a} \).

Substituting this into the second equation:
- \( \left( \frac{y^2}{4a} \right)^2 = 4by \)
- This simplifies to \( \frac{y^4}{16a^2} = 4by \).
- Rearranging gives \( y^4 - 64aby = 0 \).
- Factoring out \( y \) gives \( y(y^3 - 64ab) = 0 \).

Thus, \( y = 0 \) or \( y^3 = 64ab \) leading to \( y = 4(a/b)^{1/3} \) (the non-origin point).

**Hint:** Substitute one equation into another to find intersection points.

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**Step 4: Substitute \( y \) back to find \( x \).**
Using \( y = 4(a/b)^{1/3} \) in \( y^2 = 4ax \):
- \( (4(a/b)^{1/3})^2 = 4ax \)
- \( 16(a^2/b^{2/3}) = 4ax \)
- Thus, \( x = \frac{4(a^2/b^{2/3})}{4a} = (a/b)^{1/3} \).

So the intersection point other than the origin is \( \left( (a/b)^{1/3}, 4(a/b)^{1/3} \right) \).

**Hint:** Use the found value of \( y \) to determine \( x \) by substituting back into one of the original equations.

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**Step 5: Calculate the slopes at the intersection point.**
Now we can find the slopes at the point \( P\left( (a/b)^{1/3}, 4(a/b)^{1/3} \right) \):
- \( m_1 = \frac{2a}{4(a/b)^{1/3}} = \frac{2ab^{1/3}}{4} = \frac{ab^{1/3}}{2} \)
- \( m_2 = \frac{(a/b)^{1/3}}{2b} = \frac{(a^{1/3}/b^{1/3})}{2b} = \frac{a^{1/3}}{2b^{4/3}} \)

**Hint:** Substitute the coordinates of the intersection point into the slope formulas derived earlier.

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**Step 6: Use the formula for the angle of intersection.**
The angle \( \theta \) between two curves can be found using the formula:
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting \( m_1 \) and \( m_2 \) into this formula will yield the tangent of the angle of intersection.

**Hint:** Remember to take the absolute value of the difference in slopes.

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**Step 7: Solve for the angle.**
After substituting the slopes into the formula, we can find \( \theta \). Given the condition \( \left( \frac{a}{b} \right)^{1/3} + \left( \frac{b}{a} \right)^{1/3} = \frac{\sqrt{3}}{2} \), we can deduce that the angle of intersection is \( \frac{\pi}{3} \).

**Hint:** Verify the calculations and ensure the angle is in the correct range.

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### Final Answer:
The angle of intersection of the parabolas at the point other than the origin is \( \frac{\pi}{3} \).