Explanation:

When two regression lines are perpendicular to each other, it means that they intersect at a right angle. This occurs when the coefficient of correlation, denoted by r, is equal to 0.

The coefficient of correlation measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1. A value of +1 indicates a perfect positive linear relationship, a value of -1 indicates a perfect negative linear relationship, and a value of 0 indicates no linear relationship.

Reasoning:

To understand why the regression lines are perpendicular when r = 0, let's consider the definition of the coefficient of correlation:

r = (Σ((x - x̄)(y - ȳ))) / (sqrt(Σ((x - x̄)²) * Σ((y - ȳ)²)))

where x and y are the variables, x̄ and ȳ are their respective means, and Σ denotes summation.

When r = 0, the numerator of the equation becomes 0. This implies that the covariance between x and y is 0. In other words, there is no linear relationship between x and y.

Implication:

When there is no linear relationship between x and y, the regression lines will be perpendicular to each other. This can be understood by considering the slope of the regression lines.

The slope of the regression line for x on y is given by:

b₁ = r * (sy / sx)

where sy is the standard deviation of y and sx is the standard deviation of x.

Similarly, the slope of the regression line for y on x is given by:

b₂ = r * (sx / sy)

Since r = 0, both b₁ and b₂ become 0. This means that the regression lines have no slope and are vertical and horizontal lines, respectively. As a result, they intersect at a right angle, making them perpendicular to each other.

Therefore, the correct answer is option D) 0.