MHB How Can the Ratio k Determine the Roots of a Modified Quadratic Equation?

[anemone]

Let $k$ be the ratio of the roots of the equation $ax^2-bx+b=0$ where $a,\,b>0$.

Find the roots of the equation $\sqrt{a}x^2-\sqrt{b}x+\sqrt{a}=0$ in terms of $k$ only.

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[anemone]

Congratulations to kaliprasad for his correct solution!:)

Solution from kaliprasad:

Spoiler

Let the roots of the equation $ax^2- bx+b=0$ be $t$ and $kt$.

so $a(x-t)(x-kt) = ax^2 - bx + b = 0$
so
$(k+1)t = \frac{b}{a}$
and $kt^2 = \frac{b}{a}$

Eliminating $t$ we have $(k+1)^2/ k = \frac{b}{a}$

From the second equation $\sqrt{a}x^2- \sqrt{b}x+\sqrt{a}=0$, its product of roots = 1 and sum of roots = $\sqrt\frac{b}{a} = \frac{k+1}{\sqrt(k)}=\sqrt(k) +\frac{1}{\sqrt(k)}$

As product is 1 so the roots of the equation $\sqrt{a}x^2- \sqrt{b}x+\sqrt{a}=0$ are $\sqrt(k)$ and $\frac{1}{\sqrt(k)}$