If $12$ and $9 \frac{3}{5}$ are the geometric and harmonic means, respectively, between two numbers, find them.

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Solution

$∵ G M = \sqrt{a b} = 12$
$⟹ a b = 144$
$H M = \frac{2 a b}{a + b} = 9 \frac{3}{5} = \frac{48}{5} = \frac{2 \times 144}{a + b}$
$⟹ a + b = 30$
From the quadratic equation:
$x^{2} = S x + P$
$∵ S =$ Sum of the roots $= a + b = 30$
$P =$ Product of the roots $= a b = 144$

$∴ x^{2} - 30 x + 144 = 0$
$⟹ x^{2} - 24 x - 6 x + 144 = 0$
$⟹ x (x - 24) - 6 (x - 24) = 0$
$x = 6$ and $24$
$∴$ The required numbers are $6$ and $24$

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If 12 and 935 are the geometric and harmonic means, respectively, between two numbers, find them.

∵GM=√ab=12⟹ab=144HM=2aba+b=935=485=2×144a+b⟹a+b=30From the quadratic equation:x2=Sx+P∵S= Sum of the roots =a+b=30P= Product of the roots=ab=144∴x2−30x+144=0⟹x2−24x−6x+144=0⟹x(x−24)−6(x−24)=0x=6 and 24∴ The required numbers are 6 and 24

If $12$ and $9 \frac{3}{5}$ are the geometric and harmonic means, respectively, between two numbers, find them.