If a pair of linear equations `a_(1)x+b_(1)y+c_(1)=0` and `a_(2)x+b_(2)y+c_(2)=0`, then
(i) `(a_(1))/(a_(2))=(b_(1))/(b_(2))ne(c_(1))/(c_(2))` implies The pair of linear equations is `ul("P")`
(ii) `(a_(1))/(a_(2))ne(b_(1))/(b_(2))` implies The pair of linear equations is `ul("Q")`
(iii)`(a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))` implies The pair of linear equations is `ul("R")`

Options

A. `{:("P","Q","R"),("Inconsistent","Consistent","Consistent"):}`
B. `{:("P","Q","R"),("Inconsistent","Inconsistent","Consistent"):}`
C. `{:("P","Q","R"),("Inconsistent","Inconsistent","Inconsistent"):}`
D. `{:("P","Q","R"),("Inconsistent","Consistent","Inconsistent"):}`

Answer

To solve the given problem, we need to analyze the conditions for the pair of linear equations represented by:

1. \( a_1x + b_1y + c_1 = 0 \)
2. \( a_2x + b_2y + c_2 = 0 \)

We will determine the implications of the ratios of the coefficients.

### Step-by-Step Solution:

**Step 1: Analyze the first condition**
- Given:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
- This condition implies that the two lines represented by the equations have the same slope (since the ratios of the coefficients of \(x\) and \(y\) are equal), meaning they are parallel. However, since the ratio of the constant terms is not equal, the lines do not coincide. Therefore, they are distinct parallel lines.

**Conclusion for (i)**: The pair of linear equations is **P** (parallel lines).

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**Step 2: Analyze the second condition**
- Given:
\[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]
- This condition implies that the ratios of the coefficients of \(x\) and \(y\) are not equal. Therefore, the lines represented by the equations intersect at a unique point (since they are not parallel).

**Conclusion for (ii)**: The pair of linear equations is **Q** (intersecting lines).

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**Step 3: Analyze the third condition**
- Given:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
- This condition implies that all three ratios are equal. This means that the two lines represented by the equations are actually the same line (coincident lines).

**Conclusion for (iii)**: The pair of linear equations is **R** (coincident lines).

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### Summary of Results:
- (i) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) implies **P** (parallel lines).
- (ii) \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) implies **Q** (intersecting lines).
- (iii) \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) implies **R** (coincident lines).