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Consider the two functions $f(x)=x^2+2bx+9; \; g(x)= 3a(x+b)$, where $a,b$ are real numbers.

Each pair of $(a,b)$ may be considered to be coordinates of a point in the x-y plane. Let $S$ be the set of all such points $(a,b)$ for which the graph of $y=f(x)$ and $y=g(x)$ do not intersect.

Find the area of the region defined by S.

I understood what the question meant but I am unable to find an elegant and profound way of solving this question. Someone told me to equate $f(x)=g(x)$ and then find the value of $x$. I am not understanding why we have to equate these two. Can you explain this with an example and give a hint of starting the question?

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Hint: solve them simultaneously and apply the condition that the discriminant of this quadratic must be negative. This leads to a simple inequality in $a$ and $b$ which defines the area inside a well-known shape.

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  • $\begingroup$ ...which is a well-known shape @Jean Marie $\endgroup$ Jun 28 at 14:17

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