question 46

46. Equation of circles touching x-axis at the origin and the line 4x-3y+24=0 are
1) x2+y2-6y=0 , x2+y2+24y=0,
2) x2-y2+2y=0,  x2+y2+24=0,
3) x2+y2+18x=0, x2+y2-8x=0,
4) x2+y2+4x=0, x2+y2-16x=0,

Dear Student,
Please find below the solution to the asked query:

Let equation of cirlce be x2+y2+2gx+2fy+c=0As circle touches x-axis at 0,0, hence0+0+0+0+c=0c=0Circle touches x-axis, hence x intercept must be 02g2-c=0g2=cg=c=0Hence equation of circle becomesx2+y2+2fy=0Radius=g2+f2-c=f2=fCircle touches 4x-3y+24=0, hence perpendicular distance from centre 0,-fequal radius0-3×-f+2432+42=f3f+24=5f3f+8=5fSquare both sides9f2+64+16f=25f216f2-144f-576=0f2-9f-36=0f2-12f+3f-36=0ff-12+3f-12=0f+3f-12=0f=-3 or f=12Hence equation isx2+y2-6y=0 or x2+y2+24=0

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