Mo2men wrote:Line Q has the equation 5y - 3x = 45. If Line S is perpendicular to Q, has an integer for its y-intercept, and intersects Q in the second quadrant, then how many possible Line S's exist? (Note: Intersections on one of the axes do not count.)
(A) 25
(B) 33
(C) 36
(D) 41
(E) 58
Source: Magoosh
Line Q:
5y - 3x = 45
5y = 3x + 45
y = (3/5) + 9.
Draw line Q:
The slopes of perpendicular lines are NEGATIVE RECIPROCALS.
Since line S must be perpendicular to line Q -- and the slope of line Q is 3/5 -- the slope of line S = -5/3.
Thus, the equation for line S is as follows:
y = (-5/3)x + b.
If line S intersects line Q at the x-axis, the following figure is yielded:
Here, line S includes (-15, 0).
Plugging (-15, 0) into y = (-5/3)x + b, we get:
0 = (-5/3)(-15) + b
0 = 25 + b
b = -25.
The result is the following figure:
If line S intersects line Q at the y-axis, we get:
Since lines Q and S may not intersect at either axis, the resulting figure implies that the y-intercept of line S must be an integer value BETWEEN -25 AND 9.
Thus, the y-intercept for line S may be any integer value between -24 and 8, inclusive, for a total of 33 integer options.
The correct answer is
B.
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