If the sum of the squares of the intercepts on the axes cut off by tangent to the curve `x^(1/3)+y^(1/3)=a^(1/3),\ a >0` at `(a/8, a/8)` is 2, then `a=` 1 (b) 2 (c) 4 (d) 8

Options

A. 1
B. 2
C. 4
D. 8

Explanation

To solve the problem step by step, we will follow the given instructions and derive the value of \( a \) based on the conditions provided.

### Step 1: Understand the curve and point of tangency
The given curve is:
\[ x^{1/3} + y^{1/3} = a^{1/3} \]
We need to find the tangent to this curve at the point \( \left( \frac{a}{8}, \frac{a}{8} \right) \).

### Step 2: Differentiate the curve
To find the slope of the tangent line, we differentiate the equation of the curve with respect to \( x \):
\[ \frac{d}{dx}(x^{1/3}) + \frac{d}{dx}(y^{1/3}) = \frac{d}{dx}(a^{1/3}) \]
Using the chain rule, we get:
\[ \frac{1}{3}x^{-2/3} + \frac{1}{3}y^{-2/3} \cdot \frac{dy}{dx} = 0 \]
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{y^{2/3}}{x^{2/3}} \]

### Step 3: Evaluate the derivative at the point
Substituting \( x = \frac{a}{8} \) and \( y = \frac{a}{8} \):
\[ \frac{dy}{dx} = -\frac{\left(\frac{a}{8}\right)^{2/3}}{\left(\frac{a}{8}\right)^{2/3}} = -1 \]
Thus, the slope of the tangent at this point is \( -1 \).

### Step 4: Write the equation of the tangent line
Using the point-slope form of the line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = -1 \) and the point \( \left( \frac{a}{8}, \frac{a}{8} \right) \):
\[ y - \frac{a}{8} = -1 \left( x - \frac{a}{8} \right) \]
This simplifies to:
\[ y = -x + \frac{a}{4} \]
Rearranging gives:
\[ x + y = \frac{a}{4} \]

### Step 5: Find the intercepts
The x-intercept occurs when \( y = 0 \):
\[ x = \frac{a}{4} \]
The y-intercept occurs when \( x = 0 \):
\[ y = \frac{a}{4} \]

### Step 6: Calculate the sum of the squares of the intercepts
The sum of the squares of the intercepts is:
\[ \left(\frac{a}{4}\right)^2 + \left(\frac{a}{4}\right)^2 = 2 \left(\frac{a}{4}\right)^2 = 2 \cdot \frac{a^2}{16} = \frac{a^2}{8} \]

### Step 7: Set the equation equal to 2
According to the problem, this sum equals 2:
\[ \frac{a^2}{8} = 2 \]

### Step 8: Solve for \( a \)
Multiplying both sides by 8:
\[ a^2 = 16 \]
Taking the square root:
\[ a = 4 \quad (\text{since } a > 0) \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{4} \]