Question

Let $S$ be the set of values of paramenter $a$ for which the points of intersection of the parabolas $y^2=3ax$ and $y=\frac{1}{2}(x^2+ax+5)$ are concyclic, then $S$ can be

• A. $(-2,\infty )$
• B. $(-\infty ,2)$
• C. $(-\infty ,-2)$
• D. $(2,\infty )$

Answer 1

Answer: (C), (D)

$(2,\infty )$

Any curve througth intersection of given curves will be of form
$y^2-3ax+\lambda (x^2+ax+5-2y)=0\cdots \left(1\right)$
This should be a real circle $\Rightarrow \lambda =1$ and radius $>0$
$\Rightarrow x^2+y^2-2ax-2y+5=0\Rightarrow (x-a{)}^2+(y-1{)}^2=a^2-4$
$\Rightarrow a^2-4>0\Rightarrow a>2\text{or}a<-2$