Question

The equation of a circle with radius 5 and touching both the coordinate axes is
(a) x² + y² ± 10x ± 10y + 5 = 0
(b) x² + y² ± 10x ± 10y = 0
(c) x² + y² ± 10x ± 10y + 25 = 0
(d) x² + y² ± 10x ± 10y + 51 = 0

Answer 1

(c) x² + y² ± 10x ± 10y + 25 = 0

Case I: If the circle lies in the first 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 II: 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:

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 $x^2+y^2-10x+10y+25=0$.


Hence, the required equation of the circle is x² + y² ± 10x ± 10y + 25 = 0.