A tangent P T is drawn to the circle x²+y²=4 at the point P( 3 , 1) . A straight line L , perpendicular to P T is a tangent to the circle (x-3)²+y²-1 . Question: A common tangent of the two circles is

Step-by-step solution

From the figure it is clear that the intersection point of two direct common tangents lies on x -axis. Also P T₁ C₁ P T₂ C₂ P C₁: P C₂=2: 1 or P divides C₁ C₂ in the ratio 2: 1 externally Coordinates of P are (6,0) . Let the equation of tangent through P be y=m(x-6) Asit touches x²+y²=4 | 6 m m²+1 |=2 36 m²=4 (m²+1 ) m= 1 2 2 Equations of common tangents are y= 1 2 2 (x-6) Also x=2 is the common tangent to the two circles.