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KCET2026MathematicsQuadratic Equation

If and are acute angles such that - and + satisfy the equation ^2 - 4 + 1 = 0 , then and are respectively,

Options

  1. A45^ , 30^
  2. B30^ , 45^
  3. C30^ , 60^
  4. D60^ , 45^

Correct answer

A. 45^ , 30^

Step-by-step solution

The roots of the equation ^2 - 4 + 1 = 0 are ( - ) and ( + ) . Sum of roots: ( - ) + ( + ) = 4 Product of roots: ( - ) ( + ) = 1 Using the tangent addition formula: (2 ) = (( - ) + ( + )) = ( - ) + ( + ) 1 - ( - ) ( + ) Substituting the sum and product of roots: (2 ) = 4 1 - 1 Since is an acute angle, 2 = 90^ = 45^ . Now, substituting = 45^ into the sum of roots equation: (45^ - ) + (45^ + ) = 4 1 - 1 + + 1 + 1 - = 4 (1 - )^2 + (1 + )^2 1 - ^2 = 4 2(1 + ^2 ) 1 - ^2 = 4 1 + ^2 = 2 - 2 ^2 3 ^2 = 1 = 1 3 (since is acu

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