Show that the common tangents to the circles `x^(2)+y^(2)-6x=0andx^(2)+y^(2)+2x=0` form an equilateral triangle.

Options

A. `3sqrt(3)-(pi)/(3)`
B. `4sqrt(3)-(11pi)/(3)`
C. `4sqrt(3)-(pi)/(3)`
D. `3sqrt(3)-(11pi)/(6)`

Answer

B

Explanation

Length of DCT `(PQ)=sqrt((C_(1)C_(2))^(2)-(r_(1)-r_(2))^(2))`
`=sqrt(16-4)=2sqrt(3)`
Area of quadrilateral
`PQC_(2)C_(1)=(1)/(2)xx(1+3)xx2sqrt(3)=4sqrt(3)`
`C_(1)m=2sqrt(33)`

Applying cosine formula in `DeltaC_(1)M_(1)C_(2)`, we get
`becauseanglePC_(1)R=(2pi)/(3) & angleQC_(2)R=(pi)/(3)`
Area of circular sector `RC_(1)P=(1)/(2)xx(2pi)/(3)=(pi)/(3)`
Area of circular sector `RC_(2)Q=(1)/(2)xx(pi)/(3)xx9=(3pi)/(2)`
`because` required area
`=4sqrt(3)-(pi)/(3)-(3pi)/(2)=4sqrt(3)-(11pi)/(6)`