perplexus.info :: Just Math : Given Radical Real Product, Find the Square of Sum

Given Radical Real Product, Find the Square of Sum (Posted on 2023-05-25)

Each of x and y is a real number that satisfy this equation:

• {x+√(1+x²)}{y+√(1+y²)}= 1

Determine the value of (x+y)²

Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Solution: (Hide)

Let u = x+√(1+x²)
Substituting this in the given equation, we obtain:
u(y+√(1+ y²)=1
Or, √(1+y²)= 1/u-y
Or, 1+y²= (1/u-y)² = 1/u² -2y/u+y²
Or, 2y/u = 1/u² -1
Or, y = 1/2(1/u -u)
Now, u= x+ √(1+x²= (x-√(1+x²)/(x+√(1+x²)* (x-√(1+x²)= √(1+ x²)-x
So, 1/u = -x+ √(1+ x² ), and:
u = x+√(1+ x²)
Hence, y = 1/2(1/u-u)= 1/2(-x + √(1+ x²-(x + √(1+ x²)
=(1/2)*(-2x) = -x
=> x+y=0 => (x+y)² =0

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Solution	Brian Smith	2023-05-25 10:00:41
Spoiler?	Steve Herman	2023-05-25 09:30:22

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