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Mathematics

If b is the mean proportion between a and c, then the mean proportion between (a2 + b2) and (b2 + c2) is:

  1. a(b + c)

  2. b(a + c)

  3. c(a + b)

  4. none of these

Ratio Proportion

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Answer

Since b is the mean proportion between a and c,

⇒ab=bc⇒b2=ac.\Rightarrow \dfrac{a}{b} = \dfrac{b}{c} \\[1em] \Rightarrow b^2 = ac.

Let the required mean proportion be x. Then,

⇒a2+b2x=xb2+c2⇒x2=(a2+b2)(b2+c2).\Rightarrow \dfrac{a^2 + b^2}{x} = \dfrac{x}{b^2 + c^2} \\[1em] \Rightarrow x^2 = (a^2 + b^2)(b^2 + c^2).

Now substitute b2 = ac:

⇒ x2 = (a2 + b2)(b2 + c2)

⇒ x2 = a2b2 + a2c2 + b4 + b2c2

⇒ x2 = a2b2 + (ac)2 + b4 + b2c2

⇒ x2 = a2b2 + (b)2 + b4 + b2c2

⇒ x2 = a2b2 + b4 + b4 + b2c2

⇒ x2 = b2(a2 + 2b2 + c2)

⇒ x2 = b2(a2 + 2ac + c2)

⇒ x2 = b2(a + c)2.

Therefore,

⇒x2=b2(a+c)2⇒x=b2(a+c)2⇒x=b(a+c).\Rightarrow x^2 = b^2(a + c)^2 \\[1em] \Rightarrow x = \sqrt{b^2(a + c)^2} \\[1em] \Rightarrow x = b(a + c).

Hence, option 2 is the correct option.

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