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Two unit circles S1 and S2 have centres C1(0,0) and C2(1,0) respectively. A third unit circle S3 passes through both C1 and C2, with its centre above the x-axis. The common external tangent to S1 and S3 that does not pass through S2 has equation ax + by + 2 = 0. Match List-I with List-II: List-I: (P) The y-coordinate (ordinate) of the centre of S3 (Q) The value of a (R) The value of (a² - b) (S) The x-coordinate (abscissa) of the centre of S3 List-II: (1) 1/2 (2) sqrt(3)/2 (3) sqrt(3) (4) 4 (5) 1

1. (1) 1/2
2. (2) sqrt(3)/2
3. (3) sqrt(3)
4. (4) 4
5. (5) 1

Correct answer: (2) sqrt(3)/2

Solution

S3 passes through C1 and C2, both at distance = radius = 1 from centre of S3. Let centre = (h,k). h²+k²=1 and (h-1)²+k²=1. Subtract: h²-(h-1)²=0 => 2h-1=0 => h=1/2 (abscissa). k=sqrt(1-1/4)=sqrt(3)/2 (above x-axis). P (ordinate) = sqrt(3)/2 = List-II option (2). S (abscissa) = 1/2 = List-II option (1). For the tangent: common tangent to S1 (centre O, r=1) and S3 (centre (1/2, sqrt(3)/2), r=1) — since both have equal radii, external tangent is parallel to the line joining centres. Direction of C1C3: (1/2, sqrt(3)/2). Tangent line is perpendicular to... wait, for equal circles external tangent is parallel to the line C1C3. Normal to tangent has direction (1/2, sqrt(3)/2), i.e., direction (1, sqrt(3)). Tangent line: x + sqrt(3)*y + c = 0. Distance from C1(0,0) = |c|/sqrt(1+3) = |c|/2 = 1 => c = ±2. Tangent not passing through S2: check both options. c=2: x + sqrt(3)*y + 2 = 0. Comparing with ax+by+2=0: a=1, b=sqrt(3). Q (value of a) = 1 = option (5). R (a²-b) = 1-sqrt(3). Not in list... R = 1² - sqrt(3) = 1-1.732 = -0.732, not in list. Perhaps b is the literal coefficient value. If tangent is x+sqrt(3)*y+2=0 => a=1, b=sqrt(3). a²-b = 1-sqrt(3). Or if the tangent has c=-2 (other side): -x-sqrt(3)y+2=0 => a=-1,b=-sqrt(3); not cleaner. Let me try: distance from C3(1/2, sqrt(3)/2) to line ax+by+2=0 must = 1. And distance from C1 = 1. With a=1, b=sqrt(3): dist from (1/2,sqrt(3)/2) = |1/2 + sqrt(3)*sqrt(3)/2 + 2|/2 = |1/2+3/2+2|/2 = |4|/2 = 2 ≠ 1. So that's wrong. Reconsidering: the tangent doesn't pass through the region of S2.

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