Question

If the pair of linear equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\) represent coincident lines, then:

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

Hint:

For two linear equations to represent coincident lines, the ratios of the coefficients of \(x\), \(y\), and the constant term must be equal.

Answer 1

The Correct Option is A

For two linear equations to represent coincident lines, their ratios must be equal. The given equations are: \[ a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0 \] For the lines to coincide, the ratio of the corresponding coefficients of \(x\), \(y\), and the constant term must be the same: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \] Thus, the condition for coincident lines is that the ratios of the coefficients should be equal.