# If three distinct numbers a,b,c are in G.P.

Question:

If three distinct numbers a,b,c are in G.P. and the equations $a x^{2}+2 b x+c=0$ and $\mathrm{dx}^{2}+2 \mathrm{ex}+f=0$ have a common root, then which one of the following statements is correct?

1. $\mathrm{d}, \mathrm{e}, f$ are in A.P.

2. $\frac{\mathrm{d}}{\mathrm{a}}, \frac{\mathrm{e}}{\mathrm{b}}, \frac{f}{\mathrm{c}}$ are in G.P.

3. $\frac{\mathrm{d}}{\mathrm{a}}, \frac{\mathrm{e}}{\mathrm{b}}, \frac{f}{\mathrm{c}}$ are in A.P.

4. d,e, $f$ are in G.P.

Correct Option: , 3

Solution:

$\mathrm{a}, \mathrm{b}, \mathrm{c}$ in G.P.

say a, ar, ar $^{2}$

satisfies $a x^{2}+2 b x+c=0 \Rightarrow x=-r$

$x=-r$ is the common root, satisfies second equation $\mathrm{d}(-\mathrm{r})^{2}+2 \mathrm{e}(-\mathrm{r})+\mathrm{f}=0$

$\Rightarrow \mathrm{d} \cdot \frac{\mathrm{c}}{\mathrm{a}}-\frac{2 \mathrm{ce}}{\mathrm{b}}+\mathrm{f}=0$

$\Rightarrow \frac{\mathrm{d}}{\mathrm{a}}+\frac{\mathrm{f}}{\mathrm{c}}=\frac{2 \mathrm{e}}{\mathrm{b}}$

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