# If A(6,-1), B(1,3) and C(k,8) are three points such that AB = BC , find the value of k ?

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The direction ratios of line segment AB = ( x2 - x1 ) , ( y2 -y1)

Which can be used to represent AB as a vector .

AB = 5i -4j

BC = (k-1)i + 5j

MAGNITUDE OF AB = BC

=> 5^2 + 4^2 = k^2 + 1 -2k + 5^2

=> k^2 -2k -15 = 0

=> ( k + 3 ) ( k - 5 ) = 0

=> k = -3 or 5

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AB² =BC²

⇒((6-1)² + (-1-3)²)= ((k-1)²+ (8-3)²)

⇒(25 +16) =(k² +1- 2k +25)

⇒k² - 2k -15 =0

⇒k² -5k +3k - 15=0

⇒k(k-5) +3(k-5)= 0

⇒(k -5)(k +3)=0

(k -5) =0 or (k +3)=0

⇒