The slope of the line passing through the points `(a^(2),b)and (b^(2),a)` is

Options

A. `(1)/(a+b)`
B. `a+b`
C. `(-1)/(a+b)`
D. `-(a+b)`

Answer

To find the slope of the line passing through the points \((a^2, b)\) and \((b^2, a)\), we can use the slope formula. The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step-by-Step Solution:

1. **Identify the Points**:
The points given are \((x_1, y_1) = (a^2, b)\) and \((x_2, y_2) = (b^2, a)\).

2. **Substitute the Points into the Slope Formula**:
Substitute \(x_1, y_1, x_2, y_2\) into the slope formula:
\[ m = \frac{a - b}{b^2 - a^2} \]

3. **Factor the Denominator**:
The denominator \(b^2 - a^2\) can be factored using the difference of squares:
\[ b^2 - a^2 = (b - a)(b + a) \]

4. **Rewrite the Slope**:
Now substitute the factored form back into the slope equation:
\[ m = \frac{a - b}{(b - a)(b + a)} \]

5. **Simplify the Expression**:
Notice that \(a - b = -(b - a)\), so we can rewrite the slope as:
\[ m = \frac{-(b - a)}{(b - a)(b + a)} \]

6. **Cancel Common Terms**:
The \((b - a)\) terms in the numerator and denominator cancel out (as long as \(b \neq a\)):
\[ m = \frac{-1}{b + a} \]

### Final Result:
The slope of the line passing through the points \((a^2, b)\) and \((b^2, a)\) is:
\[ m = -\frac{1}{b + a} \]