Perpendicular lines m and n intersect at point (a, b), where a b 0

Question

Perpendicular lines m and n intersect at point (a, b), where a > b > 0. The slope of line m is between 0 and 1. Which of the following statements MUST be true ?

I. The x-intercept of line n is positive.
II. The product of the x- and y-intercepts of line m is not positive, 
III. The sum of the x-intercepts of lines m and n is positive.

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

nick1816 · Answer

Let equation of line m is y=kx+c As line m passes through (a,b), hence b=ka+c or c=b-ka Equation of line m, y=kx+(b-ka) Similarly Equation of line n, y=(-1/k)x + (b+a/k) 1. Y- intercept of line n= (b+a/k) as a,b >0 and 00 but (k-1/k)<0, when 0

IanStewart · Answer

I'd just draw pictures here to see what can happen. Our two lines meet at a point in the first quadrant (where x and y coordinates are positive), at a point (a, b), where a > b, so where the x-coordinate exceeds the y-coordinate. That means our intersection point lies below the line y=x, but above the x-axis. Line m has a slope between 0 and 1, so as it moves right, it's rising, but slowly. The perpendicular line n, therefore, is falling and quickly as it moves to the right.

So, if line n has a point on it (a, b) above the x-axis in the first quadrant, and then is falling quickly as you move right from that point, it must hit the x-axis further to the right of that point, so its x-intercept must be positive, and I must be true.

If line m has a positive slope, independent of any other information provided, the product of its x and y intercepts will be negative (or zero if it passes through the origin). You can see that just considering two cases: if its x-intercept is negative, then it rises moving right from there, so has a positive y-intercept. If its x-intercept is positive, it falls moving left from there, so its y-intercept is negative. So II must be true.

III need not be true, because the x-intercept of line m could be negative one trillion if it has a very shallow slope.

I think there's a typo in the answer choices - I imagine answer D is meant to read "I and II only", and not "I and III only". (edit - that has since been fixed in the OP)

ScottTargetTestPrep · Answer

Since the slope of line m is between 0 and 1, the slope of line n must be a negative number less than -1 since line n's slope is the negative reciprocal of the slope of m. Since lines m and n intersect at a point in the first quadrant, line n (a negatively-sloped line) that passes a point in the first quadrant must have a positive x-intercept. So Roman numeral I is true.
 
Since the slope of m is between 0 and 1 and line m passes through point (a, b), where a > b > 0 in the first quadrant, 3 cases are possible for line m:
 
1) If the y-intercept is positive, then the x-intercept is negative.
 
2) If the y-intercept is negative, then the x-intercept is positive.
 
3) If the y-intercept is 0, the x-intercept is also 0.
 
We see that in any of these 3 cases, the product is either negative or 0. So Roman numeral II is true.
 
As we can see from above, the x-intercept of line n is positive, and the x-intercept of line m is negative. However, since we don’t know the values of the two x-intercepts, we can’t tell whether the sum of the x-intercepts is positive or not.
 
Answer: D

Sourav700 · Answer

ScottTargetTestPrep   IanStewart

if you could please share your graphs here. how are we sure that y-intercept is definitely positive?

IanStewart · Answer

This is not correct, and might account for Sourav's question above. We have no way to know if the y-intercept of line m is positive or negative. If, say, line m passes through the point (2, 1), and has a slope of 1/100, then its y-intercept is definitely positive. But if line m passes through, say, (10000000, 1), and has a slope of 1/2, its y-intercept is very negative.

All we can say is what item II says: the product of the line's two intercepts is not positive. When its y-intercept is positive, its x-intercept is negative and vice versa.

CEO2021 · Answer

Attached figure may be helpful

Perpendicular lines m and n intersect at point (a, b), where a > b > 0

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