If The Two (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 Intersect In Two Distinct Points, Then

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Question

If the two (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect in two distinct points, then

Solution

Correct option is

2 < r < 8

Centres of the given circles are C1(1, 3) and C2(4, –1), and their radii,r1 = r and r2 = 3.  

We know that the two circles touch, externally if C1C­2 = r1 + r2, and internally if C1C2 = |r1 – r2|.  

Thus the two circles will cut at two distinct points if C1C2 > |r1 – r2| and C1C2 < r1 + r2, i.e., if |r1 – r2| < C1C2 < r1 + r2

  

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