Question

The value of the constant in the sphere, which passes through the three points $(1,0,0),(0,1,0),(0,0,1)$ and has radius as small as possible:

• A. $\frac{1}{3}$
• B. $\frac{-1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{-2}{3}$

Answer 1

Answer: (B)

$\frac{-1}{3}$

Let the equation of the sphere be
$(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$ [As the radius is $r$]
As it passes through $(1,0,0),(0,1,0),(0,0,1)$

So we get,
$(1-a{)}^2+(b{)}^2+(c{)}^2=r^2(a{)}^2+(1-b{)}^2+(c{)}^2=r^2(a{)}^2+(b{)}^2+(1-c{)}^2=r^2$
From these we get, $a=b=c$
So, $r^2=3a^2+1-2a$
Constant term in the sphere's equation is $3a^2-r^2$
To minimize the radius, we minimize $r^2$
So $\frac{d{r}^2}{da}=0$
So $6a-2=0$
$a=\frac{1}{3}$
[ Minimum radius is $\sqrt{23}$ ]
$3a^2-r^2=2a-1=\frac{-1}{3}$
So Option B is correct.