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JEE Main202529 Jan 2025Evening ShiftMathematicsStraight LinesActual

Let the line x+y=1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB , where O is the origin and the points M and N lie on the lines O B and A B , respectively. If the area of the triangle A M N is 4 9 of the area of the triangle O A B and AN : NB = : 1 , then the sum of all possible value(s) of is :

Options

  1. A2
  2. B5 2
  3. C1 2
  4. D13 6

Correct answer

A. 2

Step-by-step solution

aligned & Area of AOB = 1 2 & Area of AMN = 4 9 1 2 = 2 9 aligned Equation of AB is x + y =1 aligned & OA =1, AM = (45^ - ) & AN = (45^ - ) & MN = (45^ - ) aligned aligned & Ar ( AMN )= 1 2 ^2 (45^ - ) = 2 9 & =2, 1 2 & =2 is rejected & AN NB = 1 = =2 aligned

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