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To solve this problem, we need to multiply the complex numbers \( (a + bi) \) and \( (-10 + 25i) \) and then find the condition for the product to be a real number.

1. **Multiply the Complex Numbers:**
   \[
   (a + bi)(-10 + 25i) = a(-10 + 25i) + bi(-10 + 25i)
   \]
   \[
   = -10a + 25ai - 10bi + 25b i^2
   \]

2. **Simplify Using \( i^2 = -1 \):**
   \[
   = -10a + 25ai - 10bi - 25b
   \]
   \[
   = (-10a - 25b) + (25a - 10b)i
   \]

   The product can be written as a complex number \( x + yi \) where:
   \[
   x = -10a - 25b
   \]
   \[
   y = 25a - 10b
   \]

3. **Condition for the Product to be a Real Number:**
   For the product to lie on the real number line, the imaginary part \( y \) must be zero:
   \[
   25a - 10b = 0
   \]

4. **Solve for \( a \) and \( b \):**
   \[
   25a = 10b
   \]
   \[
   \frac{a}{b} = \frac{10}{25} = \frac{2}{5}
   \]

Therefore, the ratio \( a : b \) is \( 2:5 \).

Would you like a more detailed explanation of any step? 

Here are some related questions you might find helpful:
1. How do you multiply complex numbers in general?
2. What is the geometric interpretation of multiplying two complex numbers?
3. How do you determine if a complex number is purely real or purely imaginary?
4. What is the significance of the imaginary unit \( i \)?
5. How does the modulus of a complex number affect the product?

**Tip:** When multiplying complex numbers, remember to apply the distributive property and use \( i^2 = -1 \) to simplify.

Let a and b be non-zero real numbers. If the result of (a + bi)(-10 + 25i) can be represented on a real number line, find a : b.

Learn how to find the ratio a : b for the product of two complex numbers to lie on the real number line. This problem focuses on the multiplication of complex numbers and the condition that the imaginary part must be zero. Suitable for high school students.

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