Question

If a+b :✓ab = 4 : 1, then a : b= ?

Please try to solve it..​

Answer 1

Solution :-

As provided :-

$a + b : \sqrt{ab} = 4 : 1$

Question asks the Ratio of

a : b = ?

Now let us write the ratio into fraction form .

$\rightarrow \dfrac{a + b}{\sqrt{ab}} = 4$

Now we will split the the fraction :-

$\rightarrow \dfrac{a}{\sqrt{ab}} + \dfrac{a}{\sqrt{ab}} = 4$

Now we will take the whole fraction under root.

$\rightarrow \sqrt{\dfrac{a^2}{ab}} + \sqrt{\dfrac{b^2}{ab}} = 4$

$\rightarrow \sqrt{\dfrac{a}{b}} + \sqrt{\dfrac{b}{a}} = 4$

Now let $\bold{\sqrt{\frac{a}{b}} = x }$ then $\bold{\sqrt{\frac{b}{a}} = \frac{1}{x}}$

Now our Equation became :-

$\rightarrow x + \dfrac{1}{x} = 4$

$\rightarrow \dfrac{x^2 + 1}{x} = 4$

$\rightarrow \dfrac{x^2 + 1}{x} \times x = 4 \times x$

$\rightarrow x^2 + 1 = 4x$

$\rightarrow x^2 - 4x + 1 = 0$

Now we will solve the Quadratic Equation :-

x² - 4x + 1

Where

a = 1

b = -4

c = 1

Now as we have

$x = \dfrac{-b \pm \sqrt{ b^2 - 4ac}}{2a}$

By putting the values :-

$\rightarrow x = \dfrac{-(-4) \pm \sqrt{ (-4)^2 - 4(1)(1)}}{2(1)}$

$\rightarrow x = \dfrac{4 \pm \sqrt{ 16 - 4}}{2}$

$\rightarrow x = \dfrac{4 \pm \sqrt{ 12}}{2}$

$\rightarrow x = \dfrac{4\pm 2\sqrt{3}}{2}$

$\rightarrow x = 2 \pm \sqrt{3}$

But as x is a square root , hence it can not be negative

$\rightarrow x = 2 + \sqrt{3}$

Now as

$x = \sqrt{\dfrac{a}{b}} = 2 + \sqrt{3}$

So the value of

$\rightarrow \dfrac{a}{b} = \left( 2 + \sqrt{3}\right)^2$

$\rightarrow \dfrac{a}{b} = (2)^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2$

$\rightarrow \dfrac{a}{b} = 4 + 4\sqrt{3} + 3$

$\rightarrow \dfrac{a}{b} = 7 + 4 \sqrt{3}$

Hence

$\huge{\boxed{\sf{a : b = 7 + 4 \sqrt{3} : 1 }}}$

Answer 2

Step-by-step explanation:

that's helpful for you