Chapter 3 · Question 3

State the ratio test for consistency of $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ .

Answer Revealed

Direct Answer:

\frac{a_1}{a_2}\ne\frac{b_1}{b_2}

, there is one solution. If

\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}

, there is no solution. If all three ratios are equal, there are infinitely many solutions.

Simple Explanation

Compare the ratios of coefficients. Different first two ratios means intersecting lines; equal first two but different constant ratio means parallel lines; all equal means same line.

Exam-Ready Structure

For the pair

a_1x+b_1y+c_1=0

a_2x+b_2y+c_2=0

: 1.

\frac{a_1}{a_2}\ne\frac{b_1}{b_2}

gives intersecting lines and a unique solution. 2.

\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}

gives parallel lines and no solution. 3.

\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}

gives coincident lines and infinitely many solutions.

Key Points

If $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$ , there is one solution.
Use the NCERT formula or theorem carefully.
Write units and final conclusion where applicable.

State the ratio test for consistency of $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ .