Question

The range of parameter ${}^{\prime }{a}^{\prime }$ for which a unique circle will pass through the points of intersection of the rectangular hyperbola $x^2-y^2=a^2$ and the parabola $y=2x^2$, is

• A. $a\in R$
• B. $a\in (-1,1)$
• C. $a\in (-\frac{1}{2},\frac{1}{2})$
• D. $a\in (-\frac{1}{4},\frac{1}{4})$

Answer 1

The correct option is C $a\in (-\frac{1}{2},\frac{1}{2})$

Let the equation of a circle passing through the points of intersection of the hyperbola and the parabola is
$x^2-y^2-a^2+\lambda (y-2x^2)=0\Rightarrow (2\lambda -1)x^2+y^2-\lambda y+a^2=0$
For circle , $2\lambda -1=1\Rightarrow \lambda =1$
Center $(0,\frac{1}{2})$ and radius $\sqrt{\frac{1}{4}-a^2}$
$r>0\Rightarrow \frac{1}{4}-a^2>0\Rightarrow a^2<\frac{1}{4}\Rightarrow a\in (-\frac{1}{2},\frac{1}{2})$