Find the value of k if the distance between the points (2,k) and (4,3) is 8. I want a real answer not a similar answer.

Given distance between the points :
(2,k) and (4,3) is 8
so using distance formula,
√[(4-2)²+(3-k)²]=8
on squaring both sides
2²+3²-2*3*k+k²=64
4+9-6k+k =64
13-6k+k²-64. =0
k²-6k-51. =0

now solving this equation by quadratic formula,
k={-(-6)±√[(-6)²-4*1*-51]}/2
=[6 ±√(36+204)]/2
=(6 ±√240)/2
=(6 ± 4√15)/2
=[2(3± 2√15)]/2
=3± 2√15

hence the value of k is 3+(2√15) or 3-(2√15)

hope it helps!!

Points :-

A ( 2 , k )
B ( 4 , 3 ) 

AB = 8 units .

To find :- k = ?

Solution :-

By distance formula :-  
  ___________________
√ ( 4 - 2 )² + ( 3 - k )²     =   8
  ______________
√ 4 + 9 + k² - 6k      =    64

Now ,

Squaring on both sides :-

13 + k² - 6k - 64 = 0

k² - 6k - 51 = 0

By quadratic formula :-    

Suppose x = k
                           _____________
          - (- 6) +- √ (- 6)2 - 4 (- 51)
k  =  - - - - - - - - - - - - - - - - - - - - - - - - 
                            2
                 ________
=     6 +- √ 36 + 204
   -  - - - - - - - - - - - - - - 
                 2

     6 + 4√15              6 - 4√15
=  - - - - - - - -    or   - - - - - - - - - 
        2                             2

=  3 + 2√15  or  3 - 2√15


$$  ##  X = 3 + 2√15  or  3 - 2√15  ##  $$


Hope it helps ..!!