Question

The equation of a circle with radius 5 and touching both die coordinate axes is

• A. $x^2+y^2\pm 10x\pm 10y+5=0$
• B. $x^2+y^2\pm 10x\pm 10y=0$
• C. $x^2+y^2\pm 10x\pm 10y+25=0$
• D. $x^2+y^2\pm 10x\pm 10y+51=0$

Answer 1

The correct option is C

$x^2+y^2\pm 10x\pm 10y+25=0$

Case 1: If the circle lies in the quadrant:
The equation of a circle that touches both the coordinate axes and has radius a is
$x^2+y^2-2ax-2ay+a^2=0$
The given radius of the circle is 5 units, i.e. a = 5
Thus, the equation of the circle is
$x^2+y^2-10x-10y+25=0$

CaseII : If the circle lies in the second quadrant :
The equation of a circle that touches both the coordinate axes and has radius a is
$x^2+y^2-2ax-2ay+a^2=0$
The given radius of the circle is 5 units, i.e. a = 5
Thus, the equation of the circle is
$x^2+y^2+10x-10y+25=0$

Case III : If the circle lies in the third quadrant:
The equation of a circle that touches both the coordinate axes and has radius a is
$x^2+y^2+2ax+2ay+a^2=0$
The given radius of the circle is 5 units, i.e. a = 5
Thus, the equation of the circle is
$x^2+y^2-10x+10y+25=0$

Case IV : If the circle lies in the fourth quadrant :
$x^2+y^2-2ax+2ay+a^2=0$
The given radius of the circle is 5 units, i.e. a= 5
Thus, the equation of the circle is
$x^2+y^2-10x+10y+25=0$

Hence, the required equation of the circle is
$x^2+y^2\pm 1Ox\pm lOy+25=0$