Question

If $r$ is the radius of circle which touches both the coordinate axes and passes through $(-2,1),$ then which of the following is/are true?

• A. least value of $r$ is $1$.
• B. greatest value of $r$ is $6$.
• C. sum of all possible values of $r$ is $6$.
• D. there exist two circle with the given conditions.

Answer 1

Answer: (A), (C), (D)

The correct options are
A least value of $r$ is $1$.
C sum of all possible values of $r$ is $6$.
D there exist two circle with the given conditions.

Since, circle is touching both the axes and passing through $(-2,1)$, it has to lie in $2^{nd}$ quadrant.
So, the centre $=(-r,r)$, where $r$ is the radius.

Therefore, equation of circle is $(x+r{)}^2+(y-r{)}^2=r^2$
Above circle is passing through $(-2,1)$, so
$(-2+r{)}^2+(1-r{)}^2=r^2\Rightarrow r^2-4r+4+r^2-2r+1=r^2$
$\Rightarrow r^2-6r+5=0\Rightarrow (r-1)(r-5)=0\Rightarrow r=1,5$

$\therefore$ Least value of $r$ is $1$,
Greatest value of $r$ is $5$.
That means, there exist two circles.
Sum of all possible values of $r$ is $6$.