All the vertices of a rectangle are of the form (a, b) with a, b integers satisfying the equation `(a-8)^(2)-(b-7)^(2)=5`. Then the perimeter of the rectangle is

Options

A. 20
B. 22
C. 24
D. 26

Answer

A

Explanation

`(a-8)^(2)-(b-7)^(2)=5`
`(a-b-1)(a+b-15)=5`
`{:(,l_(1),l_(2)),(,"Four cases",""),(,l_(1),l_(2)),(,5,5),(,-5,-1),(,-1,-5):}`
Case-1
`a-b-1=5" & "a+b-15=1`
`rArra=11,b=5`
Case-2
`a-b-1=5" & "a+b-15=-1`
`rArra=5,b=9`
Case-3
`a-b-1=1" & "a+b-15=5`
`rArra=11,b=9`
Case-4
`a-b-1=-1" & "a+b-15=-5`
`rArr a=5,b=5`

`"Perimeter"=4+4+6+6=20`