Let P be any point on the curve ` x^(2//3)+y^(2//3)=a^(2//3). ` Then the length of the segment of the tangent between the coordinate axes in of length

Options

A. 3a
B. 4a
C. 5a
D. a

Explanation

To find the length of the segment of the tangent between the coordinate axes for the curve \( x^{2/3} + y^{2/3} = a^{2/3} \), we can follow these steps:

### Step 1: Identify the point on the curve
Let \( P(x_1, y_1) \) be a point on the curve. According to the curve's equation:
\[ x_1^{2/3} + y_1^{2/3} = a^{2/3} \]

### Step 2: Differentiate the curve equation
We differentiate the equation implicitly with respect to \( x \):
\[ \frac{d}{dx}(x^{2/3}) + \frac{d}{dx}(y^{2/3}) = 0 \]
Using the chain rule, we get:
\[ \frac{2}{3} x^{-1/3} + \frac{2}{3} y^{-1/3} \frac{dy}{dx} = 0 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Rearranging the differentiated equation gives:
\[ \frac{dy}{dx} = -\frac{y^{1/3}}{x^{1/3}} \]

### Step 4: Find the slope at point \( P(x_1, y_1) \)
Substituting \( (x_1, y_1) \) into the slope formula:
\[ \frac{dy}{dx} \bigg|_{(x_1, y_1)} = -\frac{y_1^{1/3}}{x_1^{1/3}} \]

### Step 5: Write the equation of the tangent line
Using the point-slope form of the line, the equation of the tangent line at point \( P \) is:
\[ y - y_1 = -\frac{y_1^{1/3}}{x_1^{1/3}}(x - x_1) \]
Rearranging gives:
\[ y = -\frac{y_1^{1/3}}{x_1^{1/3}} x + \left( y_1 + \frac{y_1^{1/3}}{x_1^{1/3}} x_1 \right) \]

### Step 6: Find the x-intercept (point A)
Set \( y = 0 \) to find the x-intercept:
\[ 0 = -\frac{y_1^{1/3}}{x_1^{1/3}} x + y_1 + \frac{y_1^{1/3}}{x_1^{1/3}} x_1 \]
Solving for \( x \):
\[ x = \frac{y_1 + \frac{y_1^{1/3}}{x_1^{1/3}} x_1}{\frac{y_1^{1/3}}{x_1^{1/3}}} = \frac{y_1^{1/3} x_1^{1/3}}{y_1^{1/3}} = x_1 \]

### Step 7: Find the y-intercept (point B)
Set \( x = 0 \) to find the y-intercept:
\[ y = y_1 + \frac{y_1^{1/3}}{x_1^{1/3}} x_1 \]
This gives:
\[ y = y_1 + y_1^{1/3} = y_1^{1/3}(1 + 1) = 2y_1^{1/3} \]

### Step 8: Calculate the length of the segment AB
The coordinates of points A and B are \( (x_1, 0) \) and \( (0, 2y_1^{1/3}) \) respectively. The length of segment AB can be calculated using the distance formula:
\[ AB = \sqrt{(x_1 - 0)^2 + (0 - 2y_1^{1/3})^2} = \sqrt{x_1^2 + (2y_1^{1/3})^2} \]

### Step 9: Substitute \( y_1 \) using the curve equation
From the curve equation, we know:
\[ y_1^{2/3} = a^{2/3} - x_1^{2/3} \]
Thus, substituting \( y_1 \) into the length formula gives:
\[ AB = \sqrt{x_1^2 + 4(a^{2/3} - x_1^{2/3})} \]

### Step 10: Simplify the expression
This simplifies to:
\[ AB = \sqrt{4a^{2/3}} = 2a^{1/3} \]

### Final Answer
The length of the segment of the tangent between the coordinate axes is:
\[ \text{Length} = 2a^{1/3} \]