The co-ordinates of three points are A(7, 4), B(-1, -2) and C(3t, 5-4t). Find the value of t if the three points are collinear.
If the three points are collinear, then the slope of the line passing through any two of the points should be the same as the slope of the line passing through the other two points.
Let's find the slope of the line passing through points A and B:
slope AB = (y2-y1)/(x2-x1)
= (-2-4)/(-1-7)
= -6/-8
= 3/4
Now let's find the slope of the line passing through points B and C:
slope BC = (y2-y1)/(x2-x1)
= (5-4t-(-2))/(3t-(-1))
= (7-4t)/(3t+1)
Since the three points are collinear, slope AB = slope BC:
3/4 = (7-4t)/(3t+1)
Cross-multiplying and simplifying:
12t + 4 = 28-16t
28t = 24
t = 24/28
t = 6/7
Therefore, the value of t is 6/7 if the three points A, B, and C are collinear.
To determine if the three points A(7, 4), B(-1, -2), and C(3t, 5-4t) are collinear, we can use the concept of slope.
The slope between two points (x1, y1) and (x2, y2) is calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
First, we calculate the slope between points A and B:
slope_AB = (y2 - y1) / (x2 - x1)
= (-2 - 4) / (-1 - 7)
= -6 / (-8)
= 3/4
Now, we can determine the slope between points A and C:
slope_AC = (5 - 4t - 4) / (3t - 7)
= (1 - 4t) / (3t - 7)
Since the three points A, B, and C are collinear, the slopes between any two points must be equal. Therefore, we have:
slope_AB = slope_AC
Substituting the values, we can solve for t:
3/4 = (1 - 4t) / (3t - 7)
Cross-multiplying:
3(3t - 7) = 4(1 - 4t)
9t - 21 = 4 - 16t
Combining like terms:
9t + 16t = 4 + 21
25t = 25
Simplifying:
t = 1
Therefore, the value of t that makes the three points A(7, 4), B(-1, -2), and C(3t, 5-4t) collinear is t = 1.
To determine if three points are collinear, we need to check if they lie on the same straight line.
The equation of a straight line can be written in the form y = mx + c, where m is the slope of the line and c is the y-intercept.
Let's find the equation of the line passing through points A(7, 4) and B(-1, -2):
1. Calculate the slope (m) using the slope formula:
m = (y2 - y1) / (x2 - x1)
m = (-2 - 4) / (-1 - 7)
m = -6 / -8
m = 3/4
2. Now that we have the slope, we can find the y-intercept (c) using the point-slope form:
y - y1 = m(x - x1)
-2 - 4 = (3/4)(-1 - 7)
-6 = (3/4)(-8)
-6 = -24/4
-6 = -6
Since the equation simplifies to -6 = -6, that means any value of x will satisfy the equation. Therefore, the line passes through all possible x-values.
Now, let's check if point C(3t, 5-4t) lies on the same line:
3. Plug in the coordinates of point C into the equation y = mx + c:
5 - 4t = (3/4)(3t) - 6
5 - 4t = (9/4)t - 6
4. Simplify the equation and solve for t:
Multiply both sides of the equation by 4 to get rid of the fraction:
20 - 16t = 9t - 24
Combine like terms:
-16t - 9t = -24 - 20
-25t = -44
Divide both sides by -25:
t = -44 / -25
t = 44/25
So, the value of t that makes the three points A(7, 4), B(-1, -2), and C(3t, 5-4t) collinear is t = 44/25.