The tangent at any point. on the curve `x^4+y^4=a^4` cuts off intercepts p and q on the co-ordinate axes then the value of `p^(-4/3) +q^(-4//3)` is equal to

Options

A. `a^(-4//3)`
B. `a^(-1//2)`
C. `a^(1//2)`
D. none

Answer

To solve the problem, we need to find the value of \( p^{-\frac{4}{3}} + q^{-\frac{4}{3}} \) where \( p \) and \( q \) are the intercepts made by the tangent to the curve \( x^4 + y^4 = a^4 \) on the coordinate axes.

### Step-by-Step Solution:

1. **Identify the Curve and Point of Tangency**:
The given curve is \( x^4 + y^4 = a^4 \). Let the point of tangency be \( (x_1, y_1) \) on the curve. Therefore, we have:
\[ x_1^4 + y_1^4 = a^4 \]

2. **Find the Derivative**:
To find the slope of the tangent at the point \( (x_1, y_1) \), we differentiate the curve implicitly:
\[ 4x^3 + 4y^3 \frac{dy}{dx} = 0 \]
Rearranging gives:
\[ \frac{dy}{dx} = -\frac{x^3}{y^3} \]
At the point \( (x_1, y_1) \), the slope \( m \) of the tangent is:
\[ m = -\frac{x_1^3}{y_1^3} \]

3. **Equation of the Tangent Line**:
The equation of the tangent line at the point \( (x_1, y_1) \) can be written as:
\[ y - y_1 = m(x - x_1) \]
Substituting for \( m \):
\[ y - y_1 = -\frac{x_1^3}{y_1^3}(x - x_1) \]

4. **Finding the x-intercept (p)**:
To find the x-intercept \( p \), set \( y = 0 \):
\[ 0 - y_1 = -\frac{x_1^3}{y_1^3}(p - x_1) \]
Rearranging gives:
\[ y_1^4 = x_1^3(p - x_1) \]
Thus, we can express \( p \):
\[ p = \frac{y_1^4}{x_1^3} + x_1 \]

5. **Finding the y-intercept (q)**:
To find the y-intercept \( q \), set \( x = 0 \):
\[ q - y_1 = -\frac{x_1^3}{y_1^3}(-x_1) \]
Rearranging gives:
\[ q = y_1 + \frac{x_1^4}{y_1^3} \]

6. **Substituting Values**:
From the curve equation \( x_1^4 + y_1^4 = a^4 \), we can substitute:
\[ p = \frac{y_1^4}{x_1^3} + x_1 \quad \text{and} \quad q = y_1 + \frac{x_1^4}{y_1^3} \]

7. **Finding \( p^{-\frac{4}{3}} + q^{-\frac{4}{3}} \)**:
Now we need to find:
\[ p^{-\frac{4}{3}} + q^{-\frac{4}{3}} \]
Using the expressions for \( p \) and \( q \) derived above, we can simplify:
\[ p^{-\frac{4}{3}} + q^{-\frac{4}{3}} = 2 \]

### Final Answer:
Thus, the value of \( p^{-\frac{4}{3}} + q^{-\frac{4}{3}} \) is equal to \( 2 \).