"
It's the hardest problem in the GRE official book. 90% of test takers missed it." -- per a
Manhhtan GRE blog.So congratulations to folks who got it right, and that too within a min or two.
Few coordinates behaviors to keep in mind (for quick assured solution):
for a given line:
1). Line equation is y = mx + c
2). X-intercept =\( -c/m\)
Y intercept = c
3). X & Y intercepts = 0 ( thus, m or c = 0 ), if line passes through origin
4). if given two coordinates pair for a line, ----> get line-eqn form for both, and equate them at C
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coming back to ques:
given: line doesn't pass through origin. i.e. m or c Not Equal to 0.
ques: is slope negative. i.e. is \(m < 0\) ?
A). The x-intercept of line k is twice the y-intercept of line k.\(- c/m = 2c\)
\(c(2m+1)/m = 0\)
since, c & m are not zero.
thus, \(2m + 1 = 0\)
\(m = -1/2\)
\(m < 0\)B). The product of the x-intercept and the y-intercept of line k is positive.\((-c/m) * c > 0\)
\(c^2 / m < 0\)
since, \(c^2 >= 0\)
thus,
\(m < 0\)C). Line k passes through the points and where (a, b) (r, s), where (a − r)(b − s) < 0.line eqn for 1st point --> \(b = ma + c\)
line eqn for 2nd point --> \(s = mr + c\)
since, both can be equated at C,
thus, \(b - ma = s -mr\)
\(m = (b-s) / (a-r)\)
now since, \((a − r)(b − s) < 0\).
thus, each term is opposite in sign to other.
\(m < 0\) Carcass wrote:
In the xy-plane, line k is a line that does \(not\) pass through the origin. Which of the following statements \(individually\) provide(s) sufficient additional information to determine whether the slope of line k is negative?
Indicate
all such statements.
A. The x-intercept of line k is twice the y-intercept of line k.
B. The product of the x-intercept and the y-intercept of line k is positive.
C. Line k passes through the points and where (a, b) (r, s), where (a − r)(b − s) < 0.
Practice Questions
Question: 11
Page: 340
Difficulty: medium