kindly solve if the harmonic mean and geometric mean of two positive n

1.	kindly solve if the harmonic mean and geometric mean of two positive numbers are in the ratio 4:5 then the two numbers are in the ratio ?
Answer» Answer: Numbers are in ratio 4:1 Step-by-step explanation: *Given :* The harmonic mean and geometric mean of two positive numbers are in the ratio = 4:5 Both are positive numbers *To find :* The ratio in which the numbers are Let the numbers be KA and a so, a> 0 as number is positive. *APPLY RELATION for G.M* $g.m = {(ka \times a)}^{ \frac{1}{2} } \\ \\ = a \sqrt{ka}$ *Now apply relation for H.M* $h = \frac{2(ka)a}{ka + a} \\ \\ = \frac{2(ka)a}{a(k + 1)} \\ \\ = \frac{2ka}{k + 1}$ *Now using the given Ratio relation* $\frac{h.m}{g.m} = \frac{4}{5}$ *Put the values* $\huge{ \frac{ \frac{2ka}{k + 1} }{ a\sqrt{k} } = \frac{4}{5} } \\ \\ \huge{ \frac{2ka}{(k + 1) \times (a \sqrt{k}) } = \frac{4}{5} }\\ \\ \huge{ \frac{2 \sqrt{k} }{k + 1} = \frac{4}{5} } \\ \\ \huge{5(2 \sqrt{k} ) = 4(k + 1)} \\ \\ \huge{ 10 \sqrt{k} = 4k + 4 }\\ \\$ *Now divide by 2 both sides* $5 \sqrt{k} = 2k + 2 \\ \\ 2k - 5 \sqrt{k} + 2 = 0 \\ \\$ °•° *Solve this Quadratic equation* •°• Apply Quadratic Formula $\sqrt{k} = \frac{5 \pm \sqrt{ {5}^{2} - {4}^{2} } }{2 \times 2} \\ \\ = \frac{5 \pm \sqrt{25 - 16} }{4} \\ \\ = \frac{5 \pm \sqrt{9} }{4} \\ \\ = \frac{5 \pm 3}{4}$ *Finally, we get* $\sqrt{k} = \frac{5 \pm 3}{4}$ *Two values of k are* •Taking +ve sign $\sqrt{k} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \\ \\ \sqrt{k} = 2 \\ \\ k = 4$ *Next root,* •Taking -ve sign $\sqrt{k} = \frac{5 - 3}{4} = \frac{1}{2} \\ \\ \sqrt{k} = \frac{1}{2} \\ \\ k = \frac{1}{4}$ *Now A/c to QUESTION* •°•Required Ratio $= \frac{4a}{ a } = \frac{4}{1}$ •°• The numbers are in ratio 4:1 $\rule{200}{1}$ $\boxed{\underline{\bold{\mathsf{Formula \: used}}}}$ ◾ Geometric mean = ${(a \times b) }^{ \frac{1}{2} }$ ◾Harmonic mean $= \frac{2ab}{a + b}$ ◾Quadratic formula = $\frac{b \pm \sqrt{ {b}^{2} - 4ac } }{2a}$ Where a,b c are respective co-efficients

kindly solve if the harmonic mean and geometric mean of two positive numbers are in the ratio 4:5 then the two numbers are in the ratio ?

Answer»

Answer:

Numbers are in ratio 4:1

Step-by-step explanation:

Given :

The harmonic mean and geometric mean of two positive numbers are in the ratio = 4:5

Both are positive numbers

To find : The ratio in which the numbers are

Let the numbers be KA and a

so, a> 0 as number is positive.

APPLY RELATION for G.M

$g.m = {(ka \times a)}^{ \frac{1}{2} } \\ \\ = a \sqrt{ka}$

Now apply relation for H.M

$h = \frac{2(ka)a}{ka + a} \\ \\ = \frac{2(ka)a}{a(k + 1)} \\ \\ = \frac{2ka}{k + 1}$

Now using the given Ratio relation

$\frac{h.m}{g.m} = \frac{4}{5}$

Put the values

$\huge{ \frac{ \frac{2ka}{k + 1} }{ a\sqrt{k} } = \frac{4}{5} } \\ \\ \huge{ \frac{2ka}{(k + 1) \times (a \sqrt{k}) } = \frac{4}{5} }\\ \\ \huge{ \frac{2 \sqrt{k} }{k + 1} = \frac{4}{5} } \\ \\ \huge{5(2 \sqrt{k} ) = 4(k + 1)} \\ \\ \huge{ 10 \sqrt{k} = 4k + 4 }\\ \\$

Now divide by 2 both sides

$5 \sqrt{k} = 2k + 2 \\ \\ 2k - 5 \sqrt{k} + 2 = 0 \\ \\$

°•° Solve this Quadratic equation

•°• Apply Quadratic Formula

$\sqrt{k} = \frac{5 \pm \sqrt{ {5}^{2} - {4}^{2} } }{2 \times 2} \\ \\ = \frac{5 \pm \sqrt{25 - 16} }{4} \\ \\ = \frac{5 \pm \sqrt{9} }{4} \\ \\ = \frac{5 \pm 3}{4}$