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If the points with the coordinates $(\mathrm{a}, \mathrm{ma}),\{\mathrm{b},(\mathrm{m}+1) \mathrm{b}\}$, $\{\mathrm{c},(\mathrm{m}+2) \mathrm{c}\}$ are collinear, then which one of the following is correct?
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Verified Answer
The correct answer is:
$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in harmonic progression for all $\mathrm{m}$
As given :
Points $\mathrm{A}(\mathrm{a}, \mathrm{ma}), \mathrm{B}[\mathrm{b},(\mathrm{m}+1) \mathrm{b}]$ and $\mathrm{C}[\mathrm{c},(\mathrm{m}+2) \mathrm{c}]$ are
collinear. $\Rightarrow \quad \mathrm{a}\{(\mathrm{m}+1) \mathrm{b}-(\mathrm{m}+2) \mathrm{c}\}+\mathrm{b}\{(\mathrm{m}+2) \mathrm{c}-\mathrm{ma}\}$
$+\mathrm{c}\{\mathrm{ma}-(\mathrm{m}+1) \mathrm{b}\}=0$
$\Rightarrow \quad \mathrm{mab}+\mathrm{ab}-\mathrm{mac}-2 \mathrm{ac}+\mathrm{mbc}+2 \mathrm{bc}-\mathrm{mab}+\mathrm{mac}$
$\quad-\mathrm{mbc}-\mathrm{bc}=0$
$\Rightarrow \mathrm{ab}-2 \mathrm{ac}+2 \mathrm{bc}-\mathrm{bc}=0$
$\Rightarrow \mathrm{ab}+\mathrm{bc}=2 \mathrm{ac}$
Dividing both the sides by abc, we get
$\frac{1}{c}+\frac{1}{a}=\frac{2}{b}$
$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P
$\Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c}$, are in Harmonic progression for all $\mathrm{m}$.
Points $\mathrm{A}(\mathrm{a}, \mathrm{ma}), \mathrm{B}[\mathrm{b},(\mathrm{m}+1) \mathrm{b}]$ and $\mathrm{C}[\mathrm{c},(\mathrm{m}+2) \mathrm{c}]$ are
collinear. $\Rightarrow \quad \mathrm{a}\{(\mathrm{m}+1) \mathrm{b}-(\mathrm{m}+2) \mathrm{c}\}+\mathrm{b}\{(\mathrm{m}+2) \mathrm{c}-\mathrm{ma}\}$
$+\mathrm{c}\{\mathrm{ma}-(\mathrm{m}+1) \mathrm{b}\}=0$
$\Rightarrow \quad \mathrm{mab}+\mathrm{ab}-\mathrm{mac}-2 \mathrm{ac}+\mathrm{mbc}+2 \mathrm{bc}-\mathrm{mab}+\mathrm{mac}$
$\quad-\mathrm{mbc}-\mathrm{bc}=0$
$\Rightarrow \mathrm{ab}-2 \mathrm{ac}+2 \mathrm{bc}-\mathrm{bc}=0$
$\Rightarrow \mathrm{ab}+\mathrm{bc}=2 \mathrm{ac}$
Dividing both the sides by abc, we get
$\frac{1}{c}+\frac{1}{a}=\frac{2}{b}$
$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P
$\Rightarrow \mathrm{a}, \mathrm{b}, \mathrm{c}$, are in Harmonic progression for all $\mathrm{m}$.
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