Let S 1 : x 2 + y 2 = 9 and S 2 : ( x - 2 ) 2 + y 2 = 1 . Then the locus of center of a variable circle S which touches S 1 internally and S 2 externally always passes through the points :

Step-by-step solution

∵ c 1 c 2 = r 1 - r 2 ∴ Given circles are touching internally Let a variable circle with centre P and radius r ⇒ P A = r 1 - r and P B = r 2 + r ⇒ P A + P B = r 1 + r 2 ⇒ P A + P B = 4 ( > A B ) ⇒ Locus of P is an ellipse with foci at A ( 0 , 0 ) and B ( 2 , 0 ) and length of major axis is 2 a = 4 , e = 1 2 ⇒ Centre is at ( 1 , 0 ) and b 2 = a 2 1 - e 2 = 3 if x -axis is the axis of ellipse then ⇒ E   :   ( x - 1 ) 2 4 + y 2 3 = 1 which is satisfied by 2