21. Find the no. of common tangent to x² + y² =  4  and x² + y² –  8y + 12 = 0.

Soln. The centre of  the first circle is  C₁(0, 0) and radius  r₁ = 2

And for the second circle C₂ (0, 4) and r₂ = 2.

Now,  C₁ C₂  =    =  4   and  r₁ +  r₂ =  4

∴  Circles touch each other from outside.

∴  No. of  common tangents to both the circles  =  5.

22. Find the equation of tangent to the circle x² + y² =  4  from the point  (0, 2).

Soln. Since, the point lies on the circle.

∴  The equation  of  the tangent is :  x₁x  +  y₁y  =  r²  ⇒   2y  =  4  ⇒  y  =  2.

23. Find the length of the tangent to the circle x² + y² =  4  from the point  ( 1, 7).

 Soln. Since (1, 7) lies out side the circle x² + y² =  4.

∴  The length of  the tangent   =    PT   = 

Best of Luck