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 formula for the slope of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

### Step-by-Step Solution:

1. **Identify the Points**:
We have two points:
- Point 1: \((x_1, y_1) = (a^2, b)\)
- Point 2: \((x_2, y_2) = (b^2, a)\)

2. **Substitute the Points into the Slope Formula**:
Substitute the coordinates into the slope formula:
\[ \text{slope} = \frac{a - b}{b^2 - a^2} \]

3. **Factor the Denominator**:
We can factor the denominator \(b^2 - a^2\) 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:
\[ \text{slope} = \frac{a - b}{(b - a)(b + a)} \]

5. **Simplify the Expression**:
Notice that \(a - b\) can be rewritten as \(-(b - a)\):
\[ \text{slope} = \frac{-(b - a)}{(b - a)(b + a)} \]

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

7. **Final Result**:
Thus, the slope of the line passing through the points \((a^2, b)\) and \((b^2, a)\) is:
\[ \text{slope} = -\frac{1}{a + b} \]