Question

for what values of k the points A(k+1,2k),B(3k,2k+3),C(5k-1,5k) are collinear

Answer 1

Hint: Find the equation of the line joining A and B. Since the 3 points are collinear, the point C will satisfy the equation. Put the coordinates of C in the equation, you will get your value of k.

The equation will be
y - (2k) = {(2k+3 - 2k)/(3k-(k+1))} (x - (k+1))
⇒ y - 2k = {3/(2k+1)} (x- (k+1))

Now put the point C;
5k - 2k = {3/(2k+1)} ( 5k-1 - k - 1)
⇒3k = {3(4k-2)}/(2k+1)
⇒ 6k² + 3k = 12k - 6
⇒ 6k² - 9k + 6 = 0
⇒ 2k² - 3k + 2 = 0
 
Well, this is giving complex solutions. Maybe your question is wrong.

Answer 2

Hint:-
Find the equation of line AB ..As the points A,B,C are collinear value of x and y of C must be satisfy the equation.. Then put the value of x and y in obtained equation then you will get value of K..
I hope you understand it.