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+1 vote
23.8k views
in Mathematics by (34.9k points)

Let S1 : x2 + y2 = 9 and S2 : (x – 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :

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1 Answer

+3 votes
by (34.4k points)

Correct option is (3) (2, ±\(\frac 32\))

\(\therefore\) given circle are touching internally

Let a variable circle with centre P and radius r

⇒ PA = r1 – r and PB = r2 + r 

⇒ PA + PB = r1 + r2 

⇒ PA + PB = 4 ( > AB) 

⇒ Locus of P is an ellipse with foci at A(0, 0) and B(2, 0) and length of major axis is 2a = 4, e = \(\frac 12\) 

⇒ centre is at (1, 0) and b2 = a2(1 – e2) = 3 if x-ellipse

⇒ E : \(\frac{(x-1)^2}4+\frac{y^2}3\) = 1

which is satisfied by \((2,±\frac 32)\) 

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