Question

A pair of linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) has unique solution if :

• A. \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)
• B. \(\frac{a_1}{a_2} \neq \frac{c_1}{c_2}\)
• C. \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
• D. \(\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)

Hint:

For a pair of linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\):
{Unique Solution} (lines intersect at one point): \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
{No Solution} (lines are parallel and distinct): \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
{Infinitely Many Solutions} (lines are coincident): \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) The question asks for a unique solution.

Answer 1

The Correct Option is C

Concept: The conditions for the number of solutions of a pair of linear equations in two variables, \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), are determined by the ratios of their corresponding coefficients. Conditions for Solutions: (A) Unique Solution (Intersecting Lines): The pair of linear equations has exactly one unique solution if the lines they represent intersect at a single point. This occurs when the ratio of the coefficients of \(x\) is not equal to the ratio of the coefficients of \(y\). Condition: \[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \] (B) Infinitely Many Solutions (Coincident Lines): The pair of linear equations has infinitely many solutions if the lines they represent are coincident (i.e., they are the same line). This occurs when the ratios of all corresponding coefficients are equal. Condition: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \] (C) No Solution (Parallel Distinct Lines): The pair of linear equations has no solution if the lines they represent are parallel and distinct. This occurs when the ratio of the coefficients of \(x\) is equal to the ratio of the coefficients of \(y\), but this common ratio is not equal to the ratio of the constant terms. Condition: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] Step 1: Identify the condition for a unique solution Based on the concepts above, a pair of linear equations has a unique solution if and only if: \[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \] Step 2: Compare with the given options
Option (1) \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\): This is the condition for infinitely many solutions.
Option (2) \(\frac{a_1}{a_2} \neq \frac{c_1}{c_2}\): This condition alone is not sufficient to guarantee a unique solution. For example, if \(\frac{a_1}{a_2} = \frac{b_1}{b_2}\) but \(\neq \frac{c_1}{c_2}\) (parallel lines), then \(\frac{a_1}{a_2} \neq \frac{c_1}{c_2}\) would be true, but there's no solution.
Option (3) \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\): This is the correct condition for a unique solution.
Option (4) \(\frac{b_1}{b_2} \neq \frac{c_1}{c_2}\): Similar to option (2), this condition alone is not sufficient. Therefore, the correct condition for a unique solution is \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\).