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The line ax + by = 2 (with a, b nonzero) is tangent to the circle x² + y² - 2x = 3 and is also normal to the circle x² + y² - 4y = 6. Find the values of a and b respectively.
- 1, -1
- 1, 2
- -4/3, 1
- 2, 1
Correct answer: 2, 1
Solution
The line is normal to x²+y²-4y=6 (centre (0,2)), so it passes through (0,2): 2b = 2 -> b = 1; tangency to x²+y²-2x=3 (centre (1,0), radius 2) gives distance |a-2|/sqrt(a²+1) = 2, which yields a = 2 (with b = 1).
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