full question below: In the xy-plane, line L passes through the origin and is perpendicular to the line 4x+y=k
In the xy-plane, line L passes through the origin and is perpendicular to the line 4x+y=k, where k is a constant. If the two lines intersect at the point (t,t+1), that is the value of t?
I tried a long time to figure this out. It is from a textbook so I don't think there is a problem with the question.
- 1 decade agoFavorite Answer
First off, we need the slope of the line L.
Since L is perpendicular to 4x+y=k, we find the latter's slope: m = -B/A (since the equation's in the form Ax + By = C).
m = -B/A = -4/1 = -4
L's slope should be the negative reciprocal of m: 1/4.
We know it's y-intercept is 0, since it goes through the origin (0,0).
So L's slope-intercept equation is y = (1/4)·x +0, or simply y = (1/4)·x
To solve for t, we put the point (t,t+1) into either equation. But since the second equation (4x + y = k) introduces another unneeded variable, use the one we found for L:
y = (1/4)·x
t+1 = (1/4)·t
t = -1 + t/4
(3/4)·t = -1
t = -4/3
t+1 = -1/3Source(s): Me, a math teacher.
- gateachLv 51 decade ago
First rewrite the perpendicular line in slope intercept form:
y = -4x + k
So the line L's slope must be the negative reciprocal of -4 and since it passes thru the origin it's y-intercept (b) = 0
So the equation of line L is y = (1/4)x
To find t, plug (t, t+1) in to the equation y = (1/4)x
t+1 = (1/4)t
1 = (-3/4)t
-4/3 = t
- GrampedoLv 71 decade ago
First, it's perpendicular to 4x+y=k. That means its slope is the negative reciprocal of 4x+y=k. I'll figure that out first.
Slope is -4
Slope of desired line is +1/4
My desired line looks like this:
My line passes through (0,0)
Therefore 0=(1/4)0+b, b=0
My desired line is y=1/4x
The slope of my line between(t,t+1) and (0,0) is 1/4
Therefore [(t+1)-0]/[t-0] = 1/4
3t = -4
t = -4/3
The value of t is -4/3
(-1/3) = 1/4(-4/3)
-1/3 = -1/3
Thank you for an extremely interesting problem. If I knew how to give you stars or 10 points from this stupid computer, I would!
- 1 decade ago
The original line is y = -4x+k, so has slope -4. That means L has slope 1/4. Two points on L are (0,0) and (t,t+1), so they must give the same slope: (t+1-0)/(t-0)=1/4. Now just solve for t.
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- 4 years ago
enable y=Mx be first line. the 2nd line in slope-intercept format is y=ok-4x. So M=one million/4 (sneaky, hah). we've now y=x/4 and y=ok-4x. on account that (t,t+one million) satisfies the two equations, FROM the 1st, t+one million= t/4 and 3t= -4 or t= -4/3. interior the 2nd, t+one million = ok - 4(t), from which ok=5t+one million. discover ok.