Question

The equations ax + by + c = 0 and dx + ey + c = 0 represent the same straight line if
(a) ad = be (b) ac = bd (c) bc = ad (d) ab = de

Answer 1

Generally this question has to do with the number of solutions you get from an equation like this. Of course we need at least to suppose that we can at least divide with them, so some of them is not 0. If a/d=b/e=c/f, then you get infinitely many solutions, basically for any x you will have a y as a solution. Geometrically speaking this means the two equations describe the exact same line.

Now if they describe different lines: there is only one solution or no solution, i.e. the two lines intersect in one point or if they are parallel, then there is not infinitely many solutions, so the stated relation between the parameters can not be true either.

Answer 2

The equation ax+by+c=0 and dx+ey+c=0 will represent the same line if ab=de.

• Two equations represent the same line only if coefficients are x, y, and z are proportional i.e. $\frac{a}{d}=\frac{b}{e} =\frac{c}{c} =k$.
• From the above expression we derive the condition that ae=bd.
• We also observe that k=1, i.e. a=d and b=e.
• So we can also write ab=de.
• Hence, these equations represent the same straight line if ab=de.