Question 471982: in the xy plane , line k is a line that does not pass through the origin.
which of the following statements individually provides 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 the line k is positive?
(c) line k passes through the points (a,b) and (r,s) where (a-r)(b-s)<0
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website! (a) the x intercept of line k is twice the y intercept of line k?
if so, the slope is positive because that line will go mostly through I and III quadrant
(c) line k passes through the points (a,b) and (r,s) where (a-r)(b-s)<0
the slope is 
so, this is not your answer
(b) the product of the x intercept and the y intercept of the line k is positive?
since line k is a line that does not pass through the origin, this is your answer because x intercept and the y intercept are both either positive or negative; so, their product will be positive
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! a) If the line goes through the points (2z,0) and (0,z) then the slope is z/(-2z) = -2. This works for all nonzero z, so a) works.
b) Let (x,0) and (0,y) denote the x- and y-intercepts. The slope is obviously -y/x. Given that xy is positive, then x,y are either both positive or both negative, and the quotient y/x is also positive. However the slope of k is -y/x so this is negative, so b) works.
c) The slope of the line k is (b-s)/(a-r) (the order of which we choose the points does not matter). Since (a-r)(b-s) < 0, it follows that exactly one of a-r or b-s is negative, and the slope of k is also negative (similar scenario as part b)).
Hence, all three statements work.
|
|
|
