Question

Find the radius of the circle `(xcosalpha+ysinalpha-a)^2+(xsinalpha-ycosalpha-b)^2=k^2,ifalpha` varies, the locus of its centre is again a circle. Also, find its centre and radius.

Answer 1

`x&cos^2alpha+y^2sin^2alpha+a^2+2xysinalphacosalpha-2aysinalpha-2axcosalpha+x^2sin^2alpha+y^2cos^2alpha+b^2-2xysinalphacosalpha+2bycosalpha-2bxsinalpha=k^2`
`x^2+y^2-2x(acosalpha-bsinalpha)+2y(bcosalpha-asinalpha)+a^2+b^2-k^2=0`
`r=sqrt(g^2+f^2-c)`
`r=sqrt((acosalpha+bsinalpha)^2+(bcosalpha-asinalpha)^2-a^2-b^2+k^2)`
`r=sqrt(a^2+b^2+0-a^2-b^2+k^2`
`r=k`
`x=acosalpha+bsinalpha`
`y=-(asinalpha-bcosalpha)`
`x^2+y^2=a^2cos^2alpha+2ab sinalphacosalpha+b^2sin^2alpha+a^2sin^2alpha-2ab sinalphacosalpha+b^2cos^2alpha`
`x^2+y^2=a^2+b^2`
Circle r=`sqrt(a^2+b^2`
`C=(0,0)`.