Question

find the equation of the circle which touches the axis of x at a distance 3 from the origin and intercepts a distance 6 on the axis of y.

Answer 1

let equation of circle is ----
x^2+y^2+2gx+2fy+c=0========(2)
a/c to question,
circle touch x-axis at x=3
hence,
(3)^2+0+2(3)g+0+c=0
9+6g+c=0 -------------(1)
also we know circle touch x-axis then
g^2=c
put this above equation ,
9+6g+g^2=0
(g+3)^2=0
g=-3

again ,
put this equation (1)
9-18+c=0
c=9

a/c to question ,
y-intersept =6 unit
2√(f^2-c)=6
√(f^2-9)=3
aquaring both side
f=+_3√2
now put all constant in equation (2)
then ,

x^2+y^2-6x+(+_6√2)y +9=0

Answer 2

let equation of circle be :

x^2+y^2+2gx+2fy+c=0========(2)

a/c to question,

circle touch x-axis at x=3

hence,

(3)^2+0+2(3)g+0+c=0

9+6g+c=0 -------------(1)

also we know circle touch x-axis then

g^2=c

put this above equation ,

9+6g+g^2=0

(g+3)^2=0

g=-3

again ,

put this equation (1)

9-18+c=0

c=9

a/c to question ,

y-intersept =6 unit

2√(f^2-c)=6

√(f^2-9)=3

aquaring both side

f=+_3√2

now put all constant in equation (2)

then ,

x^2+y^2-6x+(+_6√2)y +9=0