If the point A (K+1,2k), B (3K,2k+3) and C (5K-1,5K) are collinear, then find the value of K

If the point A (K+1,2k), B (3K,2k+3) and C (5K-1,5K) are collinear, th

1.	If the point A (K+1,2k), B (3K,2k+3) and C (5K-1,5K) are collinear, then find the value of K
Answer» Given points will be collinear, if area of the triangle formed by them is zero.Area =\xa0{tex}\\frac { 1 } { 2 } \\left[ x _ { 1 } \\left( y _ { 2 } - y _ { 3 } \\right) + x _ { 2 } \\left( y _ { 3 } - y _ { 1 } \\right) + x _ { 3 } \\left( y _ { 1 } - y _ { 2 } \\right) \\right]{/tex}{tex}\\Rightarrow 0 = \\frac { 1 } { 2 } {/tex}[(k + 1)(2k + 3 - 5k) + 3k(5k - 2k) + (5k - 1) (2k - 2k - 3)]{tex}\\Rightarrow {/tex}\xa00 = (k + 1)(3 - 3k) + 3k(3k) +(5k - 1)(-3){tex}\\Rightarrow {/tex}\xa00 =\xa03k - 3k2\xa0+ 3 - 3k + 9k2\xa0- 15k + 3{tex}\\Rightarrow {/tex}0 = 6k2\xa0- 15k + 6{tex}\\Rightarrow {/tex}0 = 2k2\xa0- 5k + 2{tex}\\Rightarrow {/tex}0 = 2k2\xa0- 4k - k + 2\xa0{tex}\\Rightarrow {/tex}0 = 2k(k - 2) - 1(k - 2){tex}\\Rightarrow {/tex}0 = (2k - 1)(k - 2){tex}\\Rightarrow {/tex}2k - 1 = 0 or k - 2 = 0{tex}\\Rightarrow k = \\frac { 1 } { 2 }{/tex} Or k = 2