If x-a-a-b and y-b-a-b then prove that 1-x-1-y-a2-b2-ab-

Given : x=a/a+b and y=b/a b #160; L.H.S. #10; = 1x + 1y #160; #10; = a+ba + a bb #10; = ab+b^2+a^2 abab #10; = a ...

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Addition of Rational Expressions

If x=a/a+b ...

Question

If $x = \frac{a}{a + b}$ and $y = \frac{b}{a - b}$ then prove that $\frac{1}{x} + \frac{1}{y} = \frac{a^{2} + b^{2}}{a b}$ .

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Solution

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x = 1 + a + a^{2} + . . . . .

y = 1 + b + b^{2} . . . .

| a | < 1

| b | < 1

1 + a b + a^{2} b^{2} + . . . . = \frac{x y}{x + y - 1}

\frac{x}{a} + \frac{y}{b} = 1

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{a b}{a + b}

\frac{x^{n + 1}}{a} + \frac{y^{n + 1}}{b} = {(\frac{a b}{a + b})}^{n} .

If $X = 1 + a + a^{2} + . . . . \infty$ , where |a| < 1 and $y = 1 + b + b^{2} + . . . . . \infty$ where |b| <1,

prove that $1 + a b + a^{2} b^{2} + . . . . . \infty \frac{x y}{x + y - 1}$

x = 1 + a + a^{2} + a^{3} + . . . . . \infty

y = 1 + b + b^{2} + b^{3} + . . . . . \infty

1 + a b + a^{2} b^{2} + . . . . \infty = \frac{x y}{x + y - 1}

x = 1 + a + a^{2} + a^{3} + . . . .

\infty (| a | < 1)

y = 1 + b + b^{2} + b^{3} + . . .

\infty (| b | < 1)

1 + a b + a^{2} b^{2} + a^{3} b^{3} + . . .

\infty = \frac{x y}{x + y - 1}

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