Finding Relation Between a and b for Coincident Lines

Video Explanation

Question

If the equations \(2x – 3y = 7\) and \((a+b)x – (a+b-3)y = 4a + b\) represent coincident lines, find the relation between \(a\) and \(b\).

Solution

Step 1: Write in Standard Form

\[ 2x – 3y – 7 = 0 \]

\[ (a+b)x – (a+b-3)y – (4a+b) = 0 \]

Step 2: Apply Condition for Coincident Lines

For coincident lines:

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

\[ \frac{2}{a+b} = \frac{-3}{-(a+b-3)} = \frac{-7}{-(4a+b)} \]

\[ \frac{2}{a+b} = \frac{3}{a+b-3} = \frac{7}{4a+b} \]

Step 3: Form Equations

From first two ratios:

\[ \frac{2}{a+b} = \frac{3}{a+b-3} \]

\[ 2(a+b-3) = 3(a+b) \]

\[ 2a + 2b – 6 = 3a + 3b \]

\[ a + b = -6 \quad (1) \]

From first and third ratios:

\[ \frac{2}{a+b} = \frac{7}{4a+b} \]

\[ 2(4a+b) = 7(a+b) \]

\[ 8a + 2b = 7a + 7b \]

\[ a – 5b = 0 \quad (2) \]

Step 4: Required Relation

\[ a – 5b = 0 \]

Final Answer

\[ \text{Required relation: } a – 5b = 0 \]