A variable line L passes through the point ( 3 , 5 ) and intersects the positive coordinate axes at the points A and B . The minimum area of the triangle OAB , where O is the origin, is :

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​ a x ​ + b y ​ = 1 a 3 ​ + b 5 ​ = 1 ⇒ b = a − 3 5 a ​ , a > 3 ​ A = 2 1 ​ ab = 2 1 ​ a ( a − 3 ) 5 a ​ = 2 5 ​ ⋅ a − 3 a 2 ​ ​ = 2 5 ​ ( a − 3 a 2 − 9 + 9 ​ ) = 2 5 ​ ( a + 3 + a − 3 9 ​ ) = 2 5 ​ ( a − 3 + a − 3 9 ​ + 6 ) ≥ 30 ​

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