If x = \frac{1}{2-\sqrt{3}} and y = \frac{1}{2+ \sqrt{3} }, then find the value of \frac{1}{x+1} + \frac{1}{y+1}
WBBSE
Class X
Mathematics
Quadratic Surds
03 Mark
2023
x = \frac{1}{2-\sqrt{3}}
or, x + 1 = \frac{1}{2-\sqrt{3}} + 1
= \frac{1 + 2-\sqrt{3}}{2-\sqrt{3}}
= \frac{3 -\sqrt{3}}{2-\sqrt{3}} × \frac{2 +\sqrt{3}}{2 + \sqrt{3}}
= \frac{6 + 3\sqrt{3} - 2\sqrt{3} - 3}{4 - 3}
= 3 + √3
y = \frac{1}{2+\sqrt{3}}
y + 1 = \frac{1}{2+\sqrt{3}} + 1 = 3 – √3
Now, \frac{1}{x+1} + \frac{1}{y+1}
= \frac{1}{3 + √3} + \frac{1}{3 - √3}
= \frac{3 - √3 + 3 + √3}{9 - 3}
= \frac{6}{6} = 1
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