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Question

The combined equation of three sides of a triangle is (x2y2)(2x+3y6)=0 such that the points (2,a) and (b,1) be the interior point and extrerior point of the triangle respectively. If k=[ab], then maximum value of |k| is
(where [.] denotes the greatest integer function.)

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Solution

The lines are,
y=x,y=x,2x+3y=6


When (2,a) is interior point,
Intersection point of x=2 and y=x is,
(2,2)
Intersection point of x=2 and 2x+3y=6 is,
4+3y=6y=103(2,103)
So,
2<a<103

When (b,1) is exterior point,
Intersection point of y=1 and y=x is,
(1,1)
Intersection point of y=1 and y=x is,
(1,1)
So,
b>1 or b<1

Therefore,
|k|=[ab]=4
When a103 and b1

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