The four points `A(alpha, 0), B(beta, 0), C(gamma, 0)` and `D(delta, 0)` are such that `alpha, beta` are the roots of equation `ax^(2)+2h x + b=0`, and `gamma, delta` are the roots of equation `a' x^(2)+2h' x + b'=0`. Show that the sum of the ratios in which C and D divide AB is zero, if `ab' + a' b = 2hh'`.

Answer

To solve the problem, we need to show that the sum of the ratios in which points C and D divide line segment AB is zero, given the condition \( ab' + a'b = 2hh' \).

### Step-by-Step Solution:

1. **Identify the Roots**:
- The roots of the first quadratic equation \( ax^2 + 2hx + b = 0 \) are \( \alpha \) and \( \beta \).
- The roots of the second quadratic equation \( a'x^2 + 2h'x + b' = 0 \) are \( \gamma \) and \( \delta \).

2. **Use Vieta's Formulas**:
- From Vieta's formulas, we know:
- For the first equation:
- Sum of roots: \( \alpha + \beta = -\frac{2h}{a} \)
- Product of roots: \( \alpha \beta = \frac{b}{a} \)
- For the second equation:
- Sum of roots: \( \gamma + \delta = -\frac{2h'}{a'} \)
- Product of roots: \( \gamma \delta = \frac{b'}{a'} \)

3. **Set Up the Ratios**:
- Let point C divide AB in the ratio \( \lambda:1 \). By the section formula:
\[ \gamma = \frac{\lambda \beta + \alpha}{\lambda + 1} \]
- Let point D divide AB in the ratio \( \mu:1 \):
\[ \delta = \frac{\mu \beta + \alpha}{\mu + 1} \]

4. **Express Ratios**:
- Rearranging the equations for \( \gamma \) and \( \delta \):
\[ \lambda = \frac{\alpha - \gamma}{\gamma - \beta} \]
\[ \mu = \frac{\alpha - \delta}{\delta - \beta} \]

5. **Sum of Ratios**:
- We need to show that \( \lambda + \mu = 0 \):
\[ \lambda + \mu = \frac{\alpha - \gamma}{\gamma - \beta} + \frac{\alpha - \delta}{\delta - \beta} \]

6. **Common Denominator**:
- Combine the fractions:
\[ \lambda + \mu = \frac{(\alpha - \gamma)(\delta - \beta) + (\alpha - \delta)(\gamma - \beta)}{(\gamma - \beta)(\delta - \beta)} \]

7. **Expand the Numerator**:
- Expanding the numerator:
\[ = \alpha\delta - \alpha\beta - \gamma\delta + \beta\gamma + \alpha\gamma - \beta\delta - \delta\beta + \beta^2 \]
- Simplifying gives:
\[ = \alpha\delta + \alpha\gamma + \beta\gamma + \beta\delta - 2\alpha\beta - 2\gamma\delta \]

8. **Substituting the Given Condition**:
- From the condition \( ab' + a'b = 2hh' \), we can relate the sums and products of roots:
\[ 2\gamma\delta + 2\alpha\beta = \alpha\delta + \beta\gamma + \alpha\gamma + \beta\delta \]
- Therefore, the numerator simplifies to zero:
\[ \lambda + \mu = 0 \]

### Conclusion:
Thus, we have shown that the sum of the ratios in which points C and D divide line segment AB is zero, given the condition \( ab' + a'b = 2hh' \).