Question

if the ratio of the root of equation px^2 + qx +q =0 is a:b, prove that √a/√b +√b/√a + √q/√p = o.

Answer 1

Hi Kuldeep

refer to the attachment below .

Hope it helps.

Answer 2

Answer:

The proof is below

Step-by-step explanation:

$\text{Given the ratio of the root of equation }px^2 + qx +q =0\text{ a:b}$

we have to prove

$\frac{\sqrta}{\sqrtb}+\frac{\sqrtb}{\sqrta}+\frac{\sqrtq}{\sqrtp}=0$

$\text{The equation }px^2 + qx +q =0$

Let α and β are the roots of the above equation

⇒ $\frac{\alpha}{\beta}=\frac{a}{b}$

$\frac{\sqrt{\alpha}}{\sqrt{\beta}}=\frac{\sqrta}{\sqrtb}$

$\text{sum of roots=}\alpha+\beta=\frac{-b}{a}=\frac{-q}{p}$

$\text{product of roots=}\alpha.\beta=\frac{c}{a}=\frac{q}{p}$

$\alpha+\beta+\alpha.\beta=\frac{-q}{p}+\frac{q}{p}=0$

Now taking LHS

$\frac{\sqrt a}{\sqrt b}+\frac{\sqrt b}{\sqrt a}+\frac{\sqrt q}{\sqrt p}$

$=\frac{\sqrt{\alpha}}{\sqrt{\beta}}+\frac{\sqrt{\alpha}}{\sqrt{\beta}}+\frac{1}{\alpha.\beta}$

$=0$

Hence proved