Bunuel wrote:
In the xy-coordinate plane, line L passes through the points (b, a) and (c, 0), and line M passes through the point (a, b) and the origin, where a, b, and c are different nonzero numbers. Do lines L and M intersect?
(1) \(a=(\frac{{\sqrt{5}-1}}{2})^b\)
(2) c < 0
Lines L and M will intersect if they are not parallel.
Parallel lines have EQUAL SLOPES.
Thus, the answer to the question stem will be NO if Lines L and M have equal slopes.
Question stem, rephrased:
Do Lines L and M have equal slopes?
Statement 1:Let \(b=2\), with the result that \(a=\sqrt{5}-1\)
Since Line L passes through \((2, \sqrt{5}-1)\) and \((c, 0)\), the slope of Line L = \(\frac{\sqrt{5}-1-0}{2-c} = \frac{\sqrt{5}-1}{2-c}\)
Since Line M passes through \((\sqrt{5}-1, 2)\) and \((0, 0)\), the slope of Line M = \(\frac{2-0}{\sqrt{5}-1-0} = \frac{2}{\sqrt{5}-1}\)
If the two slopes are equal, we get:
\(\frac{\sqrt{5}-1}{2-c}=\frac{2}{\sqrt{5}-1}\)
\((\sqrt{5}-1)^2 = 2(2-c)\)
\(5+1-2\sqrt{5}=4-2c\)
\(2c=2\sqrt{5}-2\)
\(c=\sqrt{5}-1=a\)
Not viable, since the prompt requires that a and c be DIFFERENT nonzero numbers.
Implication:
Since it is not viable for L and M to have equal slopes, they must have DIFFERENT slopes.
Thus, the answer to the rephrased question stem is NO.
SUFFICIENT.
Statement 2:No information about a or b.
INSUFFICIENT.
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