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Principal Axes of Ellipse

If g1,g2,g3...

Question

If $g_{1}, g_{2}, g_{3}$ are three geometric means between two positive numbers m,n then $g_{1} g_{3}$

g_{2}

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g_{2}^{2}

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2 g_{2}

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2 g_{2}^{2}

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Solution

The correct option is B $g_{2}^{2}$
They are in GP: $m, g_{1}, g_{2}, g_{3}, n$
$n = m r^{4}$
$\sqrt[4]{\frac{n}{m}} = r g_{1} = m r = m \sqrt[4]{\frac{n}{m}} g_{3} = m r^{3} = m {(\frac{n}{m})}^{3 / 4}$
$g_{1} g_{3} = m \cdot m {(\frac{n}{m})}^{1 / 4} {(\frac{n}{m})}^{3 / 4} = m^{2} \frac{n}{m} = n m (g_{2})^{2} = {(m r^{2})}^{2} = m^{2} r^{4} = m^{2} {(\frac{n}{m})}^{4 / 4} = m n g_{1} g_{3} = g_{2}^{2}$

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If g1,g2,g3 are three geometric means between two positive numbers m,n then g1g3

The correct option is B g22They are in GP: m,g1,g2,g3,nn=mr44√nm=rg1=mr=m4√nmg3=mr3=m(nm)3/4g1g3=m⋅m(nm)1/4(nm)3/4=m2nm=nm(g2)2=(mr2)2=m2r4=m2(nm)4/4=mng1g3=g22

If $g_{1}, g_{2}, g_{3}$ are three geometric means between two positive numbers m,n then $g_{1} g_{3}$