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23 May 2019, 02:21
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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
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
23 May 2019, 06:19
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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
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
23 May 2019, 08:32
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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)
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)
