I am pretty sure this question (from a university admission test exam) is wrong.

Two lines: $a_1x+b_1y+c_1=0$, $a_2x+b_2y+c_2=0$, intersect only if

(a) $a_1a_2-b_1b_2=0\;\;\;$ (b) $a_1a_2-b_1b_2\ne0\;\;\;$(c) $a_1a_2-b_1b_2=1\;\;\;$ (d) $a_1a_2-b_1b_2\ne1$

Attempted solution:

\begin{matrix} a_1x+b_1y+c_1=0& \;\; &a_2x+b_2y+c_2=0 \\ y=-\frac{a_1x}{b_1}-\frac{c_1}{b_1}&\;\; & y=-\frac{a_2x}{b_2}-\frac{c_2}{b_2}\\ \end{matrix}

If the lines are not parallel, they will always intersect. In case of non-parallel lines $$m_1\ne m_2$$ $$-\frac{a_1}{b_1}\ne-\frac{a_2}{b_2}$$ $$\require{cancel} \cancel{-}a_1b_2\ne\cancel{-}a_2b_1$$ $$a_1b_2-a_2b_1\ne0.$$

  • $\begingroup$ If you are sure then why don't you tell why? $\endgroup$
    – Thomas
    Jan 2 '15 at 6:18
  • $\begingroup$ because i cant think anything else apart from my attempted solution. $\endgroup$
    – Irtiza
    Jan 2 '15 at 6:26
  • 3
    $\begingroup$ I agree with OP, the options are all written $a_1a_2-b_1b_2$, when they should be written $a_1b_2-a_2b_1$. $\endgroup$
    – vadim123
    Jan 2 '15 at 6:28
  • $\begingroup$ Are you familiar with TEX? $\endgroup$
    – Jihad
    Jan 2 '15 at 6:29
  • 2
    $\begingroup$ i think i should learn it now $\endgroup$
    – Irtiza
    Jan 2 '15 at 6:34

If two lines do not intersect--i.e., they have no points in common--then the system of equations $$\begin{align*} a_1 x + b_1 y + c_1 &= 0 \\ a_2 x + b_2 y + c_2 &= 0 \end{align*}$$ will have no solution for $(x,y)$. Thus, if we solve the system, we find $$x = \frac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}, \quad y = \frac{a_2 c_1 - a_1 c_2}{a_1 b_2 - a_2 b_1}.$$ This solution does not exist or is indeterminate if $a_1 b_2 - a_2 b_1 = 0$.

However, some care is required: two lines coincide if $$(a_1, b_1, c_1) = k(a_2, b_2, c_2)$$ for some nonzero scalar constant $k$, and in this case, both the numerators and denominator of the aforementioned solution are zero, meaning that there are infinitely many points that the two equations share in common. So while it is a sufficient condition for $a_1 b_2 - a_2 b_1 \ne 0$ to imply that the lines intersect, it is not a strictly necessary condition, and for that reason, the question should have been better phrased by saying "two lines...intersect if..." rather than "only if"; alternatively, it might be phrased "two distinct lines...intersect only if...."

  • $\begingroup$ Actually I think one does not say that two identical lines intersect, even though they have points in common in that case. $\endgroup$ Jan 2 '15 at 14:23

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