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Product of slopes

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Product of slopes

by Nailya » Fri Apr 10, 2009 8:59 am
In the xy-coordinate plane, line l and k intersect at the point (4;3). Is the product of their slopes negative?

1) Product of the x-intercepts of lines l and k is positive.
2) Product of the y-intercepts of lines l and k is negative.

OA is C.

Is there any rules for the slopes to be negative or positive? Or should I solve this type of questions by drawing the actual lines on the plane?
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by relic » Fri Apr 10, 2009 10:11 am
This is a clever coordinate geometry problem that crams in some number properties as well.

When would the product of two slopes be negative? When one is positive and the other negative.

Statement 1 tells us that both lines cross the x-axis on the same side (positive or negative), but that doesn't give us any indication as to their slopes.

Statement 2 tells us that the lines cross the y-axis on opposite sides (one on the positive side, the other on the negative), but this information won't enable us to determine either slope.

When we consider both statements, we can determine that the lines must have slopes with opposites signs even though we cannot determine which is positive and which is negative. This is sufficient to say the product of their slopes will be negative.

A sketch is usually helpful for geometry questions, especially when other abstract ideas are mixed in.
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