Final answer:
The relationship between a and b can be found by comparing the coefficients of the quadratic equation with the equation of a straight line, y = mx + c. The slope of the lines is given by b² = a + 1, and the constant term in the equation of the lines is 2√ac = 4b.
Step-by-step explanation:
The given equation, (a + 1)x² + 4bxy + 5y² = 0, represents two parallel straight lines that are coincident, meaning they are the same line. For two lines to be coincident, their slopes must be equal, and their y-intercepts must also be equal. The relationship between a and b can be found by comparing the coefficients of the quadratic equation with the equation of a straight line, y = mx + c.
Comparing coefficients:
- The coefficient of x² in the quadratic equation is a + 1, which represents the square of the slope. The slope of a line is given by b in the equation y = mx + c. Therefore, a + 1 = b².
- The coefficient of xy in the quadratic equation is 4b, which represents the product of the slope and the constant. The constant term in the equation of the line is c. Therefore, 4b = 2√ac.
Using these relationships, we can find the relationship between a and b by solving these equations together.