Let O be the center of the circle x 2 + y 2 = r 2 , where r > 5 2 . Suppose P Q is a chord of this circle and the equation of the line passing through P and Q is 2 x + 4 y = 5 . If the center of the circumcircle of the triangle O P Q lies on the line x + 2 y = 4 , then the value of r is ______
A. A
Circumcenter C of Δ O P Q lies on x + 2 y = 4 Let C ≡ 4 - 2 α , α ∵ O C ⊥ P Q ⇒ s l o p e o f O C · s l o p e o f P Q = - 1 ⇒ α 4 - 2 α - 2 4 = - 1 ⇒ α = 8 5 ⇒ C 4 5 , 8 5 O M = | 0 + 0 - 5 | 2 2 + 4 2 = 5 2 C M = 2 4 5 + 4 8 5 - 5 2 2 + 4 2 = 3 2 5 O C = P C = 4 5 2 + 8 5 2 = 4 5 Now, P M 2 = O P 2 - O M 2 = P C 2 - C M 2 ⇒ r 2 - 5 4 = 16 5 - 9 20 ⇒ r = 2
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