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Perpendicular lines m and n intersect at point (a, b), where a > b > 0
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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 xintercept of line n is positive. II. The product of the x and yintercepts of line m is not positive, III. The sum of the xintercepts 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
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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=bka Equation of line m, y=kx+(bka)
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. Yintercept of line m= (bka) X intercept of line m= (bka)/k The product of the x and yintercepts of line m= [(bka)^2]/k Hence The product of the x and yintercepts of line m is always less than or equal to zero or can never be positive
3. X intercept of line m= (bka)/k X intercept of line n= (b+a/k)*k Sum of X intercept of line m and n= (bka)/k + (b+a/k)*k= 2a+b(k1/k) a,b>0 but (k1/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
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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 xcoordinate exceeds the ycoordinate. That means our intersection point lies below the line y=x, but above the xaxis. 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 xaxis in the first quadrant, and then is falling quickly as you move right from that point, it must hit the xaxis further to the right of that point, so its xintercept 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 xintercept is negative, then it rises moving right from there, so has a positive yintercept. If its xintercept is positive, it falls moving left from there, so its yintercept is negative. So II must be true. III need not be true, because the xintercept 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
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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 xintercept of line n is positive. II. The product of the x and yintercepts of line m is not positive, III. The sum of the xintercepts 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 negativelysloped line) that passes a point in the first quadrant must have a positive xintercept. 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 yintercept is positive, then the xintercept is negative. 2) If the yintercept is negative, then the xintercept is positive. 3) If the yintercept is 0, the xintercept 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 xintercept of line n is positive, and the xintercept of line m is negative. However, since we don’t know the values of the two xintercepts, we can’t tell whether the sum of the xintercepts is positive or not. Answer: D
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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0
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03 Jun 2019, 22:46
ScottTargetTestPrep IanStewart if you could please share your graphs here. how are we sure that yintercept is definitely positive?



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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0
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04 Jun 2019, 09:00
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 yintercept of line m must be positive, and the xintercept must be negative. Therefore, the product of the x and yintercepts 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 yintercept 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 yintercept is definitely positive. But if line m passes through, say, (10000000, 1), and has a slope of 1/2, its yintercept 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 yintercept is positive, its xintercept is negative and vice versa.
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Re: Perpendicular lines m and n intersect at point (a, b), where a > b > 0
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04 Jun 2019, 09:00






