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The chord AB is the intercept that the line y = 2x makes on the circle x² + y² - 5x = 0. If the equation of the circle with AB as diameter is x² + y² - ax - by = 0, find the value of 2a + 9b.

1. 20
2. 26
3. 50
4. 42

Correct answer: 50

Solution

Substituting y = 2x into the circle gives 5x² - 5x = 0, so x = 0 or x = 1, giving A = (0,0) and B = (1,2). The circle with AB as diameter: endpoints (0,0) and (1,2) means the equation is x(x-1) + y(y-2) = 0, i.e., x² + y² - x - 2y = 0, so a = 1, b = 2, and 2a + 9b = 2 + 18 = 20. Wait — but (0,0) is on the original circle (satisfies x²+y²-5x=0) and the new circle passes through origin so F=0. Let me verify: x(x-1)+y(y-2)=0 => x²-x+y²-2y=0 => a=1, b=2 => 2(1)+9(2)=20.

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