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), what is the value of t?
I tried drawing a picture of the problem described above, but I don't understand what I need to do to solve it.
y = m x + b
y = -4 x + k
slope, m = -4
so
slope of perpendicular = -1/-4 = .25
so form of line two is
y = .25 x + b but b = 0 because through (0,0)
then
both go through (t, t+1)
line 1
t+1 = -4 t + k so 5 t = k-1
line 2
t+1 = .25 t so .75 t = - 1
t = -1/.75 = - 1 1/3
Thank you for your help. I have one question. How did you get .75 from t+1?
Well, well, well, it seems like you're having a little trouble with your lines, my friend. Don't worry, I'm here to help you out with my clownish wisdom.
So, we've got line l passing through the origin and being perpendicular to the line 4x+y=k. The first thing we need to do is find the slope of that perpendicular line.
The slope of the perpendicular line is the negative reciprocal of the slope of the original line. The slope of the original line can be found by rearranging the equation 4x+y=k to y = -4x + k. So, the slope of the original line is -4.
To find the slope of the perpendicular line, we take the negative reciprocal of -4, which is 1/4. Are you still following me? I hope so!
Now we know that the perpendicular line has a slope of 1/4. Since line l passes through the origin, its equation can be written as y = (1/4)x.
Since the two lines intersect at the point (t, t+1), we can substitute these values into the equations of both lines to find t.
For the equation of the perpendicular line, we have (t+1) = (1/4)t. Now we can solve for t:
Multiply both sides by 4: 4(t+1) = t
Distribute: 4t + 4 = t
Subtract t from both sides: 3t + 4 = 0
Subtract 4 from both sides: 3t = -4
Divide by 3: t = -4/3
So, my dear confused friend, the value of t in this little geometry party is -4/3. I hope I was able to bring some laughter into your mathematical journey!
To solve this problem, we need to find the equation of the line that passes through the origin and is perpendicular to the line 4x+y=k. Then, we can solve for the value of t by finding the point of intersection between the two lines.
Given that line l passes through the origin (0,0), we know that the equation of line l can be written in the form y = mx, where m is the slope of the line.
Since line l is perpendicular to the line 4x+y=k, the product of their slopes will be -1. Therefore, we can find the slope of line l by taking the negative reciprocal of the slope of the line 4x+y=k.
The slope-intercept form of the line 4x+y=k is y = -4x + k. Comparing this to the general form y = mx + b, we see that the slope of the line 4x+y=k is -4.
Thus, the slope of line l is the negative reciprocal of -4, which is 1/4.
So, the equation of line l is y = (1/4)x.
Now, let's find the point of intersection between line l and the line 4x+y=k.
Since both lines intersect at the point (t,t+1), we can substitute t for x and t+1 for y in the equation of line l:
t+1 = (1/4)t
Multiply both sides of the equation by 4 to eliminate the fraction:
4(t+1) = t
Distribute:
4t + 4 = t
Subtract t from both sides of the equation:
3t + 4 = 0
Subtract 4 from both sides of the equation:
3t = -4
Divide both sides of the equation by 3:
t = -4/3
Therefore, the value of t is -4/3.
To solve this problem, we can start by finding the slope of the line 4x+y=k.
In slope-intercept form, the equation of the line 4x+y=k can be written as y = -4x + k.
Since the line l is perpendicular to this line, its slope will be the negative reciprocal of -4. Let's call the slope of line l as m.
The negative reciprocal of -4 is 1/4. So, the slope of line l is 1/4.
Since line l passes through the origin, we can use the point-slope form of a line to write its equation:
y - y1 = m(x - x1),
where (x1, y1) is a point on the line (in this case, the origin) and m is the slope.
Plugging in (0, 0) as our point and 1/4 as the slope, we have:
y - 0 = (1/4)(x - 0).
This simplifies to:
y = (1/4)x.
Now, we need to find the value of t at the point of intersection (t, t+1) for both lines.
For line l, we have:
t+1 = (1/4)t.
Simplifying this equation, we get:
4(t+1) = t.
Expanding the left side, we have:
4t + 4 = t.
Simplifying this equation further, we get:
3t = -4.
Dividing both sides by 3, we find:
t = -4/3.
Therefore, the value of t is -4/3.