f  A(0, −3), B(5, 6), C(2k + 1, −43k) are given points in which  C divides  AB in  2  :  3. Find the value of  2k.

To solve this problem, we can use the formula for the coordinates of a point that divides a line segment in a given ratio. The formula is:

C = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

where m:n is the given ratio and (x1, y1) and (x2, y2) are the coordinates of the points A and B respectively.

Given that C divides AB in the ratio 2:3, we can substitute the given values into the formula:

C = ((25 + 30)/(2+3), (26 + 3(-3))/(2+3))

Solving this gives us:

C = (2, 0)

But we are also given that C = (2k+1, -43k). Equating the two expressions for C gives us two equations:

2k + 1 = 2 -43k = 0

Solving these equations gives us k = 0.5 and k = 0 respectively. However, both values of k must be the same, so the only solution is k = 0.5.

Therefore, the value of 2k is 2*0.5 = 1.