In a graph, two lines are coincident. Which statement is correct for (a_1x+b_1y+c_1=0) and (a_2x+b_2y+c_2=0)?

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

The direct answer is option C. Coincident lines are actually the same line, so one equation can be obtained by multiplying the other by one common non-zero number. For a1x+b1y+c1=0 and a2x+b2y+c2=0, this requires a1/a2=b1/b2=c1/c2, wherever the ratios are defined. Therefore option C is correct. Option A, unequal a- and b-ratios, describes different slopes and therefore generally intersecting or non-parallel lines, not coincident lines. Option B has equal coefficient ratios but an unequal constant ratio; this is the condition for parallel, distinct lines, so the two lines never meet. Option D says b1/b2=0, which is not the general condition and would also require special coefficient values; it does not express equality of the complete equations. The important distinction is: all three ratios equal means coincident, first two equal but third unequal means parallel and distinct. Memory cue: for coincident lines, remember “all three ratios equal.”