​ 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 are in A.P. ​

Question

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.

Answer 1

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