Question

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

• A. 20
• B. 22
• C. 24
• D. 26

Answer 1

The correct option is A 20

Given :
$a,b$ are integers.
$(a-8{)}^2-(b-7{)}^2=5\Rightarrow (a-8-b+7)(a-8+b-7)=5\Rightarrow (a-b-1)(a+b-15)=5$
Let $(a-b-1)=I_1$ and $(a+b-15)=I_2$
The possible cases are,
$\begin{array}{cc}{I}_1& I_2\\ 5& 1\\ 1& 5\\ -5& -1\\ -1& -5\end{array}$

Case 1 :
$I_1=5,I_2=1\Rightarrow a-b-1=5;a+b-15=1\Rightarrow (a,b)=(11,5)$

Case 2 :
$I_1=1,I_2=5\Rightarrow a-b-1=1;a+b-15=5\Rightarrow (a,b)=(11,9)$

Case 3 :
$I_1=-5,I_2=-1\Rightarrow a-b-1=-5;a+b-15=-1\Rightarrow (a,b)=(5,9)$

Case 4 :
$I_1=-1,I_2=-5\Rightarrow a-b-1=-1;a+b-15=-5\Rightarrow (a,b)=(5,5)$


Perimeter $=4\times 2+6\times 2=20$