Question

$x^2+y^2\mp 2kx\mp 2ky+k^2$ is a set of circles. Which of the following statements are true about them?

• A. All of them touches x-axis
• B. All of them touches y-axis
• C. They are concentric
• D. The radius of all the circles are same

Answer 1

Answer: (A), (B), (D)

The correct options are
A

All of them touches x-axis

B

All of them touches y-axis

D

The radius of all the circles are same

We will go through each option and check if it is satisfied.

A) All of them touches x-axis
We know that $x2+y2+2gx+2fy+c=0$ touches x-axis if $g2=c$
In our case $g2=(\mp k)2=k2=c$
$\Rightarrow$ All of them touches x-axis

B) similarly, the condition to touch the y-axis is$f^2=c$
In our case$f=\mp k$ and $c=k^2$
$f^2=(\mp k^2)=k^2=c$
$\Rightarrow$ All of them touches y-axis also

C) They are concentric
The centre of $x^2+y^2+2gx+2fy+c=0$ is $(-g,-f)$
So the centres of $x^2+y^2\mp 2kx\mp 2ky+k^2$
Are $(k,k),(-k,k),(-k,-k)and(k,-k)$

All these have different centres

D) The radius of the $x^2+y^2+2gx+2fy+c=0$ is given by $\sqrt{g^2+f^2-c}$

In our case,y$=\sqrt{(\mp k{)}^2+(\mp k{)}^2-k^2}$

=$\sqrt{k^2}$

$=|k|$

$\Rightarrow$ Radius is same for all the circles