A family of circles depending on a parameter

Text

Consider the circles $x^2+y^2+3x-ay-3-a=0$:

• (a) find the minimum radius as $a$ varies;
• (b) decide whether the centres lie on a line, and which;
• (c) for $a=1$, find centre and radius.

Solution

The centre is $C\left(-\tfrac32;\tfrac{a}{2}\right)$ and the radius satisfies $R^2=\left(\tfrac32\right)^2+\left(\tfrac{a}{2}\right)^2-(-3-a)=\frac{a^2+4a+21}{4}.$ (a) Minimise $a^2+4a+21$: vertex at $a=-2$, value $17$. So $R_{min}=\dfrac{\sqrt{17}}{2}\approx 2{,}06$.

(b) The centre always has abscissa $-\tfrac32$: the centres lie on the line $x=-\tfrac32$.

(c) For $a=1$: $C\left(-\tfrac32;\tfrac12\right)$ and $R^2=\dfrac{26}{4}$, i.e. $R=\dfrac{\sqrt{26}}{2}\approx 2{,}55$.

$\boxed{R_{min}=\tfrac{\sqrt{17}}{2};\quad x=-\tfrac32;\quad C\left(-\tfrac32;\tfrac12\right),\ R=\tfrac{\sqrt{26}}{2}}$