Do the following equations represent a pair of coincident lines? Justify your answer.
(i) \(3x + \dfrac{1}{7}y = 3\), \(7x + 3y = 7\)
(ii) \(-2x - 3y = 1\), \(6y + 4x = -2\)
(iii) \(\dfrac{x}{2} + \dfrac{y}{5} + \dfrac{5}{16} = 0\), \(4x + 8y + \dfrac{5}{4} = 0\)
(i) No, (ii) Yes, (iii) No
Concept: Two equations represent coincident lines if and only if
\(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\).
Here, each equation is written as \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\).
(i)
First equation: \(3x + \dfrac{1}{7}y - 3 = 0\)
Second equation: \(7x + 3y - 7 = 0\)
Now, compare the ratios:
\(\dfrac{a_1}{a_2} = \dfrac{3}{7}\)
\(\dfrac{b_1}{b_2} = \dfrac{1/7}{3} = \dfrac{1}{21}\)
Since \(\dfrac{3}{7} \ne \dfrac{1}{21}\), the lines are not coincident.
(ii)
First equation: \(-2x - 3y - 1 = 0\)
Second equation: \(4x + 6y + 2 = 0\)
If we multiply the first equation by \(-2\), we get:
\(4x + 6y + 2 = 0\)
which is exactly the second equation.
Thus, all three ratios are equal, so the lines are coincident.
(iii)
First equation: \(\dfrac{x}{2} + \dfrac{y}{5} + \dfrac{5}{16} = 0\)
Multiply throughout by 80 to clear fractions:
\(40x + 16y + 25 = 0\)
Second equation: \(4x + 8y + \dfrac{5}{4} = 0\)
Multiply throughout by 4:
\(16x + 32y + 5 = 0\)
Now compare ratios:
\(\dfrac{a_1}{a_2} = \dfrac{40}{16} = 2.5\)
\(\dfrac{b_1}{b_2} = \dfrac{16}{32} = 0.5\)
Since these are not equal, the lines are not coincident.
Final Answer:
(i) No, (ii) Yes, (iii) No