Question

7. a, b, c are positive real numbers such that a2 + b2 = c2 and ab = c. Determine the value of
(a + b + c)(a + b - c)(b + c - a)(c + e - b)
b).
c2​

Answer 1

Step-by-step explanation:

Correct question:-

a, b, c are positive real numbers such that a2 + b2 = c2 and ab = c. Determine the value of

(a + b + c)(a + b - c)(b + c - a)(c + a - b)

Given:-

a, b, c are positive real numbers such that a2 + b2 = c2 and ab = c.

To find:-

Determine the value of (a + b + c) (a + b - c) (b + c - a)(c + a - b).

Solution:-

Given that:-

a,b,c are the positive real numbers

a²+b²=c²-----(1)

ab=c----------(2)

Now,

(a + b + c) (a + b - c) (b + c - a)(c + a - b)

=>[(a+b+c)(a+b-c)](b+c-a)(c+a-b)

(since (a+b)(a-b)=a²-b²)

Here, a=(a+b); b=c

=>[(a+b)²-c²][(b + c - a)(c + a - b)]

=>(a²+2ab+b²-c²)(bc+ba-b²+c²+ac-bc-ac-a²+ab)

=>(a²+b²-c²+2ab)(2ab-a²-b²+c²)

=>(a²+b²-c²+2ab)[2ab-(a²+b²)+c²]

=>(c²-c²+2ab)(2ab-c²+c²)

(from(1))

=>(0+2ab)(2ab+0)

=>(2ab)(2ab)

=>4(ab)²

=>4c² (from(2))

Answer:-

The value of

(a + b + c) (a + b - c) (b + c - a)(c + a - b)=4c²

Using formulae:-

• (a+b)(a-b)=a²-b²