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Questions
If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are proportional.
If b is the mean proportional between a and c, prove that a, c, a² + b², and b² + c² are in proportion.
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Solution
∵ b is the mean proportional between a and c, then,
b² = a × c
⇒ b² = ac ...(i)
Now a, c, a2 + b2 and b2 + c2 are in proportion
⇒ `a/c = (a^2 + b^2)/(b^2 + c^2)`
R.H.S.:
`(a^2 + b^2)/(b^2 + c^2)`
Substitute b2 = ac
`(a^2 + b^2)/(b^2 + c^2)`
= `(a^2 + ac)/(ac + c^2)`
= `(a(a + c))/(c(a + c))`
= `a/c`
∴ `a/c = (a^2 + b^2)/(b^2 + c^2)`
Hence proved.
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