Question

If x=5-√3/5+√3 and y=5+√3/5-√3 show that x²-y²=-10√3/11

Answer 1

Hope it will helps uh...!!!

Answer 2

Answer:

Given: $x=\frac{5-\sqrt{3}}{5+\sqrt{3}}\:\:and\;\:y=\frac{5+\sqrt{3}}{5-\sqrt{3}}$

To Show : $x^2-y^2=\frac{-10\sqrt{3}}{11}$

Consider,

$x=\frac{5-\sqrt{3}}{5+\sqrt{3}}$

$x=\frac{5-\sqrt{3}}{5+\sqrt{3}}\times\frac{5-\sqrt{3}}{5-\sqrt{3}}$

$x=\frac{25+3-10\sqrt{3}}{25-3}$

$x=\frac{14-5\sqrt{3}}{11}$

Now Consider,

$y=\frac{5+\sqrt{3}}{5-\sqrt{3}}$

$y=\frac{5+\sqrt{3}}{5-\sqrt{3}}\times\frac{5+\sqrt{3}}{5+\sqrt{3}}$

$y=\frac{25+3+10\sqrt{3}}{25-3}$

$y=\frac{14+5\sqrt{3}}{11}$

Now,

$LHS=x^2-y^2$

$=(\frac{14-5\sqrt{3}}{11})^2-(\frac{14+5\sqrt{3}}{11})^2$

$=\frac{196+75-140\sqrt{3}}{121}-\frac{196+75+140\sqrt{3}}{121}$

$=\frac{-280\sqrt{3}}{121}$

$=\frac{-10\sqrt{3}}{11}$

$=RHS$

Hence Proved