For what values of a and b will be equations 2x + 3y = 7, (a – b) x

1.	For what values of a and b will be equations 2x + 3y = 7, (a – b) x + (a + b)y = (3a + b – 2) represent coincident lines ? A) a = -5, b = 1 B) a = 5, b = 1 C) a = -5, b = -1 D) a = 5, b = -1
Answer» Correct option is (B) a = 5, b = 1 Conditions for coincident lines is \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) \(\Rightarrow\) \(\frac{2}{a-b}=\frac{3}{a+b}=\frac{-7}{-(3a+b-2)}\) \(\Rightarrow\) \(\frac{2}{a-b}=\frac{3}{a+b}\) and \(\frac2{a-b}=\frac{7}{(3a+b-2)}\) \(\Rightarrow\) 2 (a+b) = 3 (a - b) and 2 (3a + b - 2) = 7 (a - b) \(\Rightarrow\) 2a + 2b = 3a - 3b \(\Rightarrow\) 2a - 3a = -3b - 2b \(\Rightarrow\) -a = -5b \(\Rightarrow\) a = 5b ____________(1) and 2 (3a + b - 2) = 7 (a - b) \(\Rightarrow\) 6a + 2b - 4 = 7a - 7b \(\Rightarrow\) 6a - 7a + 2b + 7b - 4 = 0 \(\Rightarrow\) -a + 9b - 4 = 0 \(\Rightarrow\) -5b + 9b - 4 = 0 (From (1)) \(\Rightarrow\) 4b - 4 = 0 \(\Rightarrow\) b = \(\frac44\) = 1 \(\therefore a=5\times1=5\) (From (1)) Hence, a = 5 & b = 1 Correct option is B) a = 5, b = 1

For what values of a and b will be equations 2x + 3y = 7, (a – b) x + (a + b)y = (3a + b – 2) represent coincident lines ? A) a = -5, b = 1 B) a = 5, b = 1 C) a = -5, b = -1 D) a = 5, b = -1

Answer»

Correct option is (B) a = 5, b = 1

Conditions for coincident lines is

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

\(\Rightarrow\) \(\frac{2}{a-b}=\frac{3}{a+b}=\frac{-7}{-(3a+b-2)}\)

\(\Rightarrow\) \(\frac{2}{a-b}=\frac{3}{a+b}\) and \(\frac2{a-b}=\frac{7}{(3a+b-2)}\)

\(\Rightarrow\) 2 (a+b) = 3 (a - b)

and 2 (3a + b - 2) = 7 (a - b)

\(\Rightarrow\) 2a + 2b = 3a - 3b

\(\Rightarrow\) 2a - 3a = -3b - 2b

\(\Rightarrow\) -a = -5b

\(\Rightarrow\) a = 5b ____________(1)

and 2 (3a + b - 2) = 7 (a - b)

\(\Rightarrow\) 6a + 2b - 4 = 7a - 7b

\(\Rightarrow\) 6a - 7a + 2b + 7b - 4 = 0

\(\Rightarrow\) -a + 9b - 4 = 0

\(\Rightarrow\) -5b + 9b - 4 = 0 (From (1))

\(\Rightarrow\) 4b - 4 = 0

\(\Rightarrow\) b = \(\frac44\) = 1

\(\therefore a=5\times1=5\) (From (1))

Hence, a = 5 & b = 1

Correct option is B) a = 5, b = 1