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

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

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23 May 2019, 02:21
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27% (03:03) correct 73% (02:40) wrong based on 51 sessions

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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

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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

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23 May 2019, 06:19
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 0<k<1
hence b+a/k will always positive

2. Y-intercept of line m= (b-ka)
X- intercept of line m= -(b-ka)/k
The product of the x- and y-intercepts of line m= -[(b-ka)^2]/k
Hence The product of the x- and y-intercepts of line m is always less than or equal to zero or can never be positive

3. X- intercept of line m= -(b-ka)/k
X- intercept of line n= (b+a/k)*k
Sum of X- intercept of line m and n= -(b-ka)/k + (b+a/k)*k= 2a+b(k-1/k)
a,b>0 but (k-1/k)<0, when 0<k<1
Hence Sum of X- intercept of line m and n can be positive or negative, depending upon the values of a,b and k

IMO D
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Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

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23 May 2019, 08:32
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)
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Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

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Updated on: 05 Jun 2019, 06:50
Bunuel wrote:
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

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.

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Originally posted by ScottTargetTestPrep on 27 May 2019, 19:00.
Last edited by ScottTargetTestPrep on 05 Jun 2019, 06:50, edited 1 time in total.
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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

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03 Jun 2019, 22:46
ScottTargetTestPrep IanStewart

if you could please share your graphs here. how are we sure that y-intercept is definitely positive?
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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0  [#permalink]

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04 Jun 2019, 09:00
1
ScottTargetTestPrep wrote:
Since the slope of m is between 0 and 1 and it passes through point (a, b), where a > b > 0 in the first quadrant, the y-intercept of line m must be positive, and the x-intercept must be negative. Therefore, the product of the x- and y-intercepts of line m is negative. So Roman numeral II is true.

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.
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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0   [#permalink] 04 Jun 2019, 09:00
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