The lines of regression of y on x is `a_(1)x+b_(1)y+c_(1)=0` and that of x on y is `a_(2)x+b_(2)y+c_(2)=0`, then

Options

A. `a_(1)b_(2)lea_(2)b_(1)`
B. `a_(1)a_(2)leb_(1)b_(2)`
C. `a_(2)b_(1)lea_(1)b_(2)`
D. none of these

Answer

To solve the problem regarding the lines of regression of y on x and x on y, we will follow these steps:

### Step 1: Write the equations of the regression lines
The equations of the regression lines are given as:
1. For y on x: \( a_1 x + b_1 y + c_1 = 0 \)
2. For x on y: \( a_2 x + b_2 y + c_2 = 0 \)

### Step 2: Rearrange the equation for y on x
From the equation of the regression line of y on x, we can express y in terms of x:
\[ b_1 y = -a_1 x - c_1 \implies y = -\frac{a_1}{b_1} x - \frac{c_1}{b_1} \]
Thus, the regression coefficient \( b_{yx} \) (slope of y on x) is:
\[ b_{yx} = -\frac{a_1}{b_1} \]

### Step 3: Rearrange the equation for x on y
From the equation of the regression line of x on y, we can express x in terms of y:
\[ a_2 x = -b_2 y - c_2 \implies x = -\frac{b_2}{a_2} y - \frac{c_2}{a_2} \]
Thus, the regression coefficient \( b_{xy} \) (slope of x on y) is:
\[ b_{xy} = -\frac{b_2}{a_2} \]

### Step 4: Use the property of regression coefficients
We know that the product of the two regression coefficients is less than or equal to 1:
\[ b_{yx} \cdot b_{xy} \leq 1 \]
Substituting the values we found:
\[ \left(-\frac{a_1}{b_1}\right) \cdot \left(-\frac{b_2}{a_2}\right) \leq 1 \]
This simplifies to:
\[ \frac{a_1 b_2}{b_1 a_2} \leq 1 \]

### Step 5: Rearranging the inequality
Rearranging the above inequality gives:
\[ a_1 b_2 \leq a_2 b_1 \]

### Conclusion
Thus, the relationship we derived is:
\[ a_1 b_2 \leq a_2 b_1 \]
This corresponds to the first option given in the question.

### Final Answer
The correct option is:
**a1 b2 ≤ a2 b1**

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