Question

(X-a) /(x-b) +(x-b) /(x-a) =a/b+b/a
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Answer 1

(X-a) /(x-b) +(x-b) /(x-a)

=((x-a)^2+(x-b)^2)/((x-a)(x-b))

=((x^2+a^2-2xa)+(x^2+b^2-2bx))/(x^2-(a+b)x+ab)

=(2x^2-2x(a+b)+a^2+b^2)/(x^2-(a+b)x+ab)

now

a/b+b/a= (a^2+b^2)/ab

equating both equations

(2x^2-2x(a+b)+a^2+b^2)/(x^2-(a+b)x+ab)=(a^2+b^2)/ab

(2x^2-2x(a+b)+a^2+b^2)(ab)=(x^2-(a+b)x+ab)(a^2+b^2)

(2ab)x^2-2x(ab)(a+b)+(a^2+b^2)(ab)=x^2(a^2+b^2)-(a+b)(a^2+b^2)x+(ab)(a^2+b^2)

(2ab)(x^2)-2x(a^2b+b^2a)=(a^2+b^2)x^2-(a+b)(a^2+b^2)x

2abx^2-2xa^2b-2xb^2a=a^2x^2+b^2x^2-a^3x-b^3x-a^2bx-b^2ax

x^2(a^2+b^2-2ab)=x(a^3+b^3+a^2b+b^2a)

x(a-b)^2=(a+b)(a^2+b^2)

x=((a+b)(a^2+b^2))/(a-b)(a-b)

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Answer 2

Answer:

(x-a) / (x-b) +(x-b) / (x-a) = a/b +b/a

(x-a)²+(x-b)² / (x-a)(x-b) =a²+b² / ab

x² +a²-2xa +x²+b²-2xb / x²-xa-xb+ab = a²+b² / ab

2x²+a²+b²-2xa-2xb × ab = a²+b² ×(x²-xa-xb+ab)

2x²+a²+b²-2xa-2xb-a²-b² = 0×(x²-xa-xb+ab) /ab

2x²-2xa-2xb = 0/ab

2x(x-a-b) = 0/ab

x-a-b = 0/2xab

x = a+b / 2xab

x² = a+b /2ab