Question

Find x from the equation:(a+x+√a^2-x^2):(a+x-√a^2-x^2)=b:x​

Answer 1

Given : (a+x+√a²-x²):(a+x-√a²-x²)=b:x​

To find : x

Solution:

a+x+√a²-x²  = √(a + x)²  + √(a +x )(a - x)  =   √(a +x ) (√(a + x) + √(a -x ))

a+x-√a²-x²  = √(a + x)²  - √(a +x )(a - x)  =    √(a +x ) (√(a + x) -  √(a -x )

(a+x+√a²-x²):(a+x-√a²-x²) = √(a +x ) (√(a + x) + √(a -x )) / √(a +x ) (√(a + x) -  √(a -x )

=> (a+x+√a²-x²):(a+x-√a²-x²) =  (√(a + x) + √(a -x )) / (√(a + x) -  √(a -x )

Rationalizing

= (a + x + a - x + 2√(a + x)√(a -x )) / ( (a + x) - (a - x)

= (2a + 2√(a² - x²))/ 2x

= (a +  √(a² - x²) /  x

(a +  √(a² - x²) /  x  = b/x

=> a +  √(a² - x²) = b

=> √(a² - x²)  = b - a

Squaring both sides

=> a² - x² = b² + a² - 2ab

=> - x² = b²   - 2ab

=>  x² = 2ab -  b²

=>   x =  ±√( 2ab -  b²)

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