Question

find the value of k if the points A ( K + 1, 2k ) b (3K, 2k + 3 ) and C ( 5K - 1, 5K ) are collinear

Answer 1

$\underline{ \large{ \text{ANSWER}}} : \\ \\ \text{The given points are} \\ \text{A (k + 1, 2k), B (3k, 2k + 3) and C (5k - 1, 5k)} \\ \\ \text{Since the points be collinear,} \\ \text{the slope of} \: \: \overline{ \text{AB}} = \text{the slope of} \: \: \overline{ \text{BC}}$

$\implies \it{ \frac{3k - (k + 1)}{(2k + 3) - 2k} = \frac{(5k - 1) - 3k}{5k - (2k + 3)} } \\ \\ \implies \it{ \frac{3k - k - 1}{2k + 3 - 2k} = \frac{5k - 1 - 3k}{5k - 2k - 3} } \\ \\ \implies \it{ \frac{2k - 1}{3} = \frac{2k - 1}{3k - 3} } \\ \\ \implies \it{ \frac{2k - 1}{ \cancel{3}} = \frac{2k - 1}{ \cancel{3}(k - 1)} } \\ \\ \implies \it{(k - 1)(2k - 1) - (2k - 1) = 0} \\ \\ \implies \it{(2k - 1)(k - 1 - 1) = 0} \\ \\ \implies \it{(2k - 1)(k - 2) = 0}$

$\therefore \text{either 2k - 1 = 0, \: or \: k - 2 = 0} \\ \\ \to \it{k = \frac{1}{2} } \: \: \: \text{and} \: \: \: \it{k = 2} \\ \\ \implies \boxed{ \bold{k = \frac{1}{2}, \: 2 }} \\ \\ \bigstar \: \underline{ \large{ \text{MarkAsBrainliest}}} \: \bigstar$