Question

Assertion :Let $a$ and $b$ be length of the legs of a right triangle with following properties
(a) All three sides of the triangle are integer.
(b) The perimeter of triangle is numerically equal to the area of the triangle, it is given that $a<b$.

The number of ordered pairs $(a,b)$ is equal to $2$.

Reason: Maximum possible perimeter of the triangle is $30$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$a+b+\sqrt{a^2+b^2}=\left(1/2\right)ab$
$\Rightarrow (2a+2b-ab{)}^2=4(a^2+b^2)$
$\Rightarrow ab+8-4a-4b=0$
$\Rightarrow (a-4)(b-4)=8$
$\Rightarrow a-4=1,b-4=8$ or $a-4=2,b-4=4$ (since a,b are integers)
Ordered pairs are (5, 12) or (6, 8)
So statement 1 is true
Perimeter is maximum when $(a,b)=(5,12)$
and is equal to $5+12+\sqrt{5^2+{12}^2}=30$
so statement 2 is also true. But statement-1 does not follow from statement-2