# 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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2015-12-22T12:14:35+05:30
When two points are given i.e A (x1 , y1 ) , B(x2 , y2 )
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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2015-12-22T14:43:07+05:30

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By using distance formula,
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
⇒ k= 5 or  -3