Try this beautiful Problem on Algebra based on Problem on Curve from AMC 10 A, 2018. You may use sequential hints to solve the problem.
Which of the following describes the set of values of $a$ for which the curves $x^{2}+y^{2}=a^{2}$ and $y=x^{2}-a$ in the real $x y$ -plane intersect at
exactly 3 points?
Algebra
greatest integer
Pre College Mathematics
AMC-10A, 2018 Problem-14
$a>\frac{1}{2}$
We have to find out the value of \(a\)
Given that $y=x^{2}-a$ . now if we Substitute this value in $x^{2}+y^{2}=a^{2}$ we will get a quadratic equation of $x$ and \(a\). if you solve this equation you will get the value of \(a\)
Now can you finish the problem?
After substituting we will get $x^{2}+\left(x^{2}-a\right)^{2}$=$a^{2} \Longrightarrow x^{2}+x^{4}-2 a x^{2}=0 \Longrightarrow x^{2}\left(x^{2}-(2 a-1)\right)=0$
therefore we can say that either \(x^2=0\Rightarrow x=0\) or \(x^2-(2a-1)=0\)
\(\Rightarrow x=\pm \sqrt {2a-1}\). Therefore
Now Can you finish the Problem?
Therefore \(\sqrt {2a-1} > 0\)
\(\Rightarrow a>\frac{1}{2}\)
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