Question

If from any point `P` on the circle `x^2+y^2+2gx+2fy+c=0`, tangents are drawn to the circle `x^2+y^2+2gx+2fy+csin^2 alpha+(g^2+f^2)cos^2 alpha=0`, then the angle between the tangents is :
(A) `alpha`
(B) `2 alpha`
(C) `alpha/2`
(D) `alpha/3`

Answer 1

P(h,k) to the cords
`x^2+y^2+2xy+2fy=0`
`T=h^2+k^2+2gh+2fk+c`
`T=h^2+k^2+2gh+2fk+sin^alpha+(g^2+f^2)cos^2alpha`
`sintheta(h,k)` lies on the circle `x^2+y^2+2gx+2fy+c=0`
`t^2=(h^2+k^2gh+2fk+c)+csin^alpha+(g^2+f^2)cos^2alpha-c`
`=(g^2+f^2)cos^2alpha-c(1-sin^2alpha)`
`r^2=g^2f^2-(csin^2alpha+(g^2+f^2)cos^2alpha)`
`=(g^2+f^2-c)sin^2alpha`
`tantheta=r/T=sqrt((g^2+f^2-c)sin^2alpha)/sqrt(g^2+f^2-c cos^2alpha)`
`sqrt(sin^2alpha/cos^2alpha)=tanalpha`
`theta=alpha`
option b is coorect