Question

If

a : b = c : d

Then, prove that


$\frac{ \sqrt{ {a}^{2} + {c}^{2} } }{ \sqrt{ {b}^{2} + {d}^{2} } } = \frac{ \sqrt{ac + \frac{ {c}^{3} }{a} } }{ \sqrt{bd + \frac{ {d}^{3} }{b} } }$
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Answer 1

Answer:

Given :

a / b = c / d

Let a / b = c / d = k

a = b k

c = d k

$\sqrt{\dfrac{a^2+c^2}{b^2+d^2}}\\\\\implies \sqrt{\dfrac{b^2k^2+d^2k^2}{b^2+d^2}}\\\\\implies \sqrt{\dfrac{k^2(b^2+d^2)}{b^2+d^2}}\\\\\implies \sqrt{k^2}\\\\\implies k$

$\sqrt{\dfrac{ac+\dfrac{c^3}{a}}{bd+\dfrac{d^3}{b}}}\\\\\implies \sqrt{\dfrac{bdk^2+\dfrac{d^3k^3}{bk}}{bd+\dfrac{d^3}{b}}}\\\\\implies \sqrt{\dfrac{bdk^2+\dfrac{d^3k^2}{b}}{bd+\dfrac{d^3}{b}}}\\\\\implies \sqrt{k^2\dfrac{(bd+\dfrac{d^3}{b})}{bd+\dfrac{d^3}{b}}}}\\\\\implies \sqrt{k^2}\\\\\implies k$

LHS = RHS

Hence proved !

Step-by-step explanation:

Substitute the values and then solve .

Both RHS and LHS will be equal to k which is same as a:b or c:d .

Hence the given equation is proved !

Answer 2

Solution:-

Refer to the Attachment!!