Question

The point (k+1,1),(2k+1,3)and(2k+2,2k) are collinear find the value of k

Answer 1

if points (a,b), (c,d), (e,f) are colinear then
(d-b)/(c-a)=(f-b)/(e-a)
so here (3-1)/(2k+1-k-1)=(2k-1)/(2k+2-k-1)
or 2/k=(2k-1)/(k+1)
or 2k+2=2k^2-k
or 2k^2-3k-2=0
or 2k^2-(4-1)k-2=0
or 2k^2-4k+k-2=0
or 2k (k-2)+1 (k-2)=0
or (k-2)(2k+1)=0
or K=either 2 or (-1/2)

Answer 2

Given:

The point (k+1,1),(2k+1,3)and(2k+2,2k) are collinear

To Find:

The value of k

Solution:

If the three points are collinear then we can say that the slope from points 1 to 3 will be the same as the slope from points 1 to 2 because they lie in the same line, we can also say that the area enclosed by the three points will be used and we can find the area enclosed by three points using the matrix formula but here we will go for slope method to find the value of k

The formula to find the slope between points (x1,y1) and (x2,y2) will be

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

using this formula lets equate the slope of points (k+1,1) (2k+1,3) and (k+1,1) (2k+2,2k) which goes as

$\frac{3-1}{2k+1-k-1} =\frac{2k-1}{2k+2-k-1} \\2k+2=2k^2-k\\2k^2-3k-2=0$

now using the quadratic formula to find the value of k,

$k=\frac{3\pm \sqrt{9+16} }{4}\\=2,-\frac{1}{2}$

Hence, the value of k can be either 2 or -1/2.