Ratio and Proportion — Mathematics STD 10 — Question
ICSE BoardEnglish MediumSTD 10MathematicsRatio and Proportion3 Marks
Question
If a, b, c are in continued proportion, prove that $a : c = (a^2 + b^2) : (b^2 + c^2).$
✓
Answer
$a , b$ and $c$ are the continued proportion
$a: b=b: c$
$\Rightarrow \frac{a}{b}=\frac{b}{c} $
$\Rightarrow b ^2= ac$
Now $\frac{a}{c}=\frac{a^2+b^2}{b^2+c^2}$
$=a\left(b^2+c^2\right)=c\left(a^2+b^2\right)$
$\text{L.H.S.}$
$\Rightarrow a\left(b^2+c^2\right) $
$\Rightarrow a\left(a c+c^2\right) $
$\Rightarrow a c(a+c)$
$\text { R.H.S. } $
$\Rightarrow c \left( a ^2+ b ^2\right) $
$\Rightarrow c \left( a ^2+ ac \right) $
$\Rightarrow ac ( a + c ) $
$\text { L.H.S. = R.H.S. }$
Hence proved.
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