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If p, q, r are in G.P, and the equations\(p^2 + 2qx + r = 0\) and \(dx^2 + 2ex + f = 0\) have a common ront, then show that \(\frac{d}{p},\) \(\frac{e}{q},\) \(\frac{f}{r}\) are in A.P. |
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Answer» It is given that p,q,r are in G.P. ∴ q2 + pr Now, px2 + 2qx + r = 0 ⇒ px2 + 2\(\sqrt{prx}\) + r = 0 ⇒ \((\sqrt{px} +\sqrt{r})^2\) = 0 ⇒ \((\sqrt{px} +\sqrt{r})\) = 0 ⇒ x = \(-\sqrt{\frac{r}{p}}\) It is given that the equations px2, + 2px + r = 0 and dx2 + 2ex + f − 0 have a common root and the equation px2, + 2qx + r = 0 has equal roots equal to −\(\sqrt{\frac{r}{p}}\) ∴ −\(\sqrt{\frac{r}{p}}\) is a root of the equation dx2 + 2ex + f = 0 ⇒ \(d\frac{r}{p} − 2e\sqrt{\frac{r}{p}} + f\) = 0 ⇒ \(\frac{d}{p} − 2e\sqrt{\frac{1}{pe}} + \frac{f}{r}\) [Dividing though out by r] ⇒ \(\frac{d}{p} − \frac{2e}{q} +\frac{f}{r}\) = 0 [∴ q2 = pr] ⇒ 2\(\frac{e}{q}\) = \(\frac{d}{p}\) + \(\frac{f}{r}\) ⇒ \(\frac{d}{p}\), \(\frac{e}{q}\), \(\frac{f}{r}\) are in A. P Hence proved |
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