If the distance between the points …

Distance between the points P(x, 2) and Q(3, - 6) is 10.
Using distance formula PQ = 10
{tex}\Rightarrow{/tex} {tex}\sqrt { ( x - 3 ) ^ { 2 } + \{ 2 - ( - 6 ) \} ^ { 2 } }{/tex} = 10
{tex}\Rightarrow{/tex} {tex}\sqrt { x ^ { 2 } + 3 ^ { 2 } - 2 \times 3 \times x + ( 2 + 6 ) ^ { 2 } }{/tex} = 10
{tex}\Rightarrow{/tex} {tex}\sqrt { x ^ { 2 } + 9 - 6 x + 8 ^ { 2 } }{/tex} = 10
{tex}\Rightarrow{/tex} {tex}\sqrt { x ^ { 2 } + 9 - 6 x + 64 }{/tex} = 10
{tex}\Rightarrow{/tex} {tex}\sqrt { x ^ { 2 } - 6 x + 73 }{/tex} = 10
Squaring both sides, we get
x² - 6x + 73 = 100
{tex}\Rightarrow{/tex} x² - 6x + 73 - 100 = 0
{tex}\Rightarrow{/tex} x²- 6x -27 = 0
{tex}\Rightarrow{/tex} x²- 9x + 3x-2 7 = 0
{tex}\Rightarrow{/tex} x(x - 9) + 3(x - 9) = 0
{tex}\Rightarrow{/tex}  (x - 9) (x + 3) = 0
{tex}\Rightarrow{/tex} either x-9 = 0 or x + 3 = 0
{tex}\Rightarrow{/tex} x = 9 or x = - 3
Ignoring x = - 3 as it is given that x is a positive integer.
Thus, only solution is x = 9.