At a point `(a// 8, a// 8)` on the curve `x^(1//3) + y^(1//3) = a^(1//3)` (a>0) tangent is drawn. If the portion of the tangent intercepted betweenthe axes be of length `sqrt2` then a=

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the slope of the tangent line
We start with the curve given by the equation:

\[ x^{1/3} + y^{1/3} = a^{1/3} \]

To find the slope of the tangent line at the point \((\frac{a}{8}, \frac{a}{8})\), we differentiate the curve implicitly with respect to \(x\).

Differentiating both sides:

\[ \frac{1}{3} x^{-2/3} + \frac{1}{3} y^{-2/3} \frac{dy}{dx} = 0 \]

Rearranging gives:

\[ \frac{dy}{dx} = -\frac{y^{2/3}}{x^{2/3}} \]

### Step 2: Calculate the slope at the point \((\frac{a}{8}, \frac{a}{8})\)
Substituting \(x = \frac{a}{8}\) and \(y = \frac{a}{8}\) into the slope formula:

\[ \frac{dy}{dx} = -\frac{(\frac{a}{8})^{2/3}}{(\frac{a}{8})^{2/3}} = -1 \]

### Step 3: Write the equation of the tangent line
Using the point-slope form of the equation of a line, we have:

\[ y - \frac{a}{8} = -1 \left(x - \frac{a}{8}\right) \]

Simplifying this gives:

\[ y - \frac{a}{8} = -x + \frac{a}{8} \]

Rearranging, we find:

\[ y + x = \frac{a}{4} \]

### Step 4: Find the intercepts with the axes
To find the x-intercept, set \(y = 0\):

\[ 0 + x = \frac{a}{4} \implies x = \frac{a}{4} \]

To find the y-intercept, set \(x = 0\):

\[ y + 0 = \frac{a}{4} \implies y = \frac{a}{4} \]

### Step 5: Calculate the length of the intercepted segment
The length of the segment intercepted between the axes is given by the distance formula between the points \((\frac{a}{4}, 0)\) and \((0, \frac{a}{4})\):

\[ \text{Length} = \sqrt{\left(\frac{a}{4} - 0\right)^2 + \left(0 - \frac{a}{4}\right)^2} = \sqrt{\left(\frac{a}{4}\right)^2 + \left(\frac{a}{4}\right)^2} \]

This simplifies to:

\[ \text{Length} = \sqrt{2 \left(\frac{a}{4}\right)^2} = \sqrt{2} \cdot \frac{a}{4} = \frac{a \sqrt{2}}{4} \]

### Step 6: Set the length equal to \(\sqrt{2}\)
According to the problem, this length is equal to \(\sqrt{2}\):

\[ \frac{a \sqrt{2}}{4} = \sqrt{2} \]

### Step 7: Solve for \(a\)
Multiplying both sides by 4:

\[ a \sqrt{2} = 4 \sqrt{2} \]

Dividing both sides by \(\sqrt{2}\):

\[ a = 4 \]

### Conclusion
The value of \(a\) is:

\[ \boxed{4} \]