For what value of k will make the three points of (1, 4), (-3, 16) and (k, -2) collinear ?
Collinear points lie on the same straight line.
Slope of straight line:
m = ( y1 - y2 ) / ( x1 - x2 )
In this case:
x1 = 1 , y1 = 4 , x2 = - 3 , y2 = 16
m = ( y1 - y2 ) / ( x1 - x2 ) =
( 4 - 16 ) / [ 1 - ( - 3 ) ] =
- 12 / ( 1 + 3 ) = - 12 / 4 = - 3
m = - 3
also:
m = ( y2 - y3 ) / ( x2 - x3 )
In this case:
x2 = - 3 , y2 = 16 , x3 = k , y3 = - 2
m = - 3 = ( y2 - y3 ) / ( x2 - x3 ) =
[ 16 - ( - 2 ) ] / ( - 3 - k ) =
( 16 + 2) / ( - 3 - k ) = 18 / ( - 3 - k )
- 3 = 18 / ( - 3 - k )
Cross multiply
( - 3 ) * ( - 3 ) - 3 * ( - k ) = 18
9 + 3 k = 18
3 k = 18 - 9 = 9
k = 9 / 3 = 3
k = 3
(1,4), (-3,16), (k,-2).
m1 = (16-4)/(-3-1) = 12/-4 = -3.
The slope should be -3 at all points:
m2 = (-2-16)/k-(-3)) = (-2-16/(k+3) = -18/(k+3) = -3,
-18/(k+3) = -3,
-3k - 9 = -18,
K = 3.
To determine the value of k that will make the three points collinear, we can use the slope-intercept form of a linear equation.
Step 1: Find the slope of the line passing through the first two points.
The slope (m) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Given points (1, 4) and (-3, 16), we have:
m = (16 - 4) / (-3 - 1)
m = 12 / -4
m = -3
Step 2: Write the linear equation using the slope-intercept form:
y = mx + b
Using point (1, 4):
4 = -3(1) + b
4 = -3 + b
b = 7
The equation of the line passing through (1, 4) and (-3, 16) is:
y = -3x + 7
Step 3: Substitute the coordinates of the third point (k, -2) into the equation and solve for k.
-2 = -3k + 7
-3k = -2 - 7
-3k = -9
k = -9 / -3
k = 3
Therefore, the value of k that will make the three points collinear is k = 3.
To determine the value of k that will make the three points collinear, we need to check if the slope between any two pairs of points is equal. If the slopes are equal, then the points are collinear.
The slope between two points (x1, y1) and (x2, y2) is given by the formula: slope = (y2 - y1) / (x2 - x1)
Let's check the slopes between the given points (1, 4) and (-3, 16) as well as between (-3, 16) and (k, -2):
Slope between (1, 4) and (-3, 16):
slope1 = (16 - 4) / (-3 - 1) = 12 / (-4) = -3
Slope between (-3, 16) and (k, -2):
slope2 = (-2 - 16) / (k - (-3)) = -18 / (k + 3)
Since we want the points to be collinear, the slopes must be equal. Therefore, we have:
-3 = -18 / (k + 3)
We can solve this equation for k by cross-multiplying:
-3 * (k + 3) = -18
Simplifying:
-3k - 9 = -18
Adding 9 to both sides:
-3k = -9
Dividing by -3:
k = 3
Therefore, the value of k that will make the points (1, 4), (-3, 16), and (k, -2) collinear is k = 3.