Question

1) If x = √ 7 + √3/√7 - √3 and xy = 1 let us show that, x2+xy+y2/x2-xy+y2 =12/11​

Answer 1

Answer:

Given :-

• x = $\dfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}$ and xy = 1.

Prove That :-

• $\dfrac{x² + xy + y²}{x² - xy + y²} =\: \dfrac{12}{11}$

Solution :-

$\because$ $x =\: \dfrac{√7 + √3}{√7 - √3} and xy =\: 1$

Now,

➛ $y =\: \dfrac{1}{x} =\: \dfrac{√7 - √3}{√7 + √3}$

Then,

⇒ $x + y =\: \dfrac{√7 + √3}{√7 - √3} + \dfrac{√7 - √3}{√7 + √3}$

⇒ $\dfrac{(√7 + √3)² + (√7 - √3)²}{(√7 - √3)(√7 + √3)}$

⇒ $\dfrac{(7 + 3 + 2√7) + (7 + 3 - 2 × √7 × √3)}{(√7)² - (√3)²}$

⇒ $\dfrac{20}{7 - 3}$

⇒ $\dfrac{20}{4}$

⇒ 5

Now,

↦ $\dfrac{x² + xy + y²}{x² - xy + y²}$

↦ $\dfrac{(x + y)² - xy}{(x + y)² - 3xy}$

↦ $\dfrac{(√5)² - 1}{(5)² - 3 × 1}$

↦ $\dfrac{25 - 1}{25 - 3}$

↦ $\dfrac{24}{22}$

➡ $\dfrac{12}{11}$

Hence, Proved

Answer 2

Step-by-step explanation:

Answer :-

25 - 1/ 25 - 3

24/22

12/11

proved:- 12/11