# Selina Solutions Class 9 Mathematics Solutions for Exercise 1(D) in Chapter 1 - Chapter 1- Rational and Irrational Numbers

Question 2 Exercise 1(D)

Simplify:

\frac{\sqrt{x^{2}+y^{2}}-y}{x-\sqrt{x^{2}-y^{2}}} \div \frac{\sqrt{x^{2}-y^{2}}+x}{\sqrt{x^{2}+y^{2}}+y}

\begin{aligned} &\begin{aligned} & \frac{\sqrt{x^{2}+y^{2}}-y}{x-\sqrt{x^{2}-y^{2}}} \div \frac{\sqrt{x^{2}-y^{2}}+x}{\sqrt{x^{2}+y^{2}}+y} \\ =& \frac{\sqrt{x^{2}+y^{2}-y}}{x-\sqrt{x^{2}-y^{2}}} \times \frac{\sqrt{x^{2}+y^{2}}+y}{\sqrt{x^{2}-y^{2}}+x} \\ -\frac{\sqrt{x^{2}+y^{2}-y}}{x-\sqrt{x^{2}-y^{2}}} \times \frac{\sqrt{x^{2}+y^{2}+y}}{x+\sqrt{x^{2}-y^{2}}} \end{aligned}\\ &\text { Using the identify }\\ &(a+b)(a-b)=a^{2}-b^{2}\\ &\begin{array}{l} \text { we get. } \\ \qquad \begin{array}{l} \left(\sqrt{x^{2}+y^{2}}\right)^{2}-y^{2} \\ x^{2}-\left(\sqrt{x^{2}-y^{2}}\right)^{2} \\ x^{2}+y^{2}-y^{2} \\ \frac{x^{2}}{x^{2}}-x^{2}+y^{2} \\ =\frac{x^{2}}{y^{2}} \end{array} \end{array} \end{aligned}

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