MHT CET202520 Apr 2025Morning ShiftMathematicsPair of LinesActual
The acute angle between the diagonals of a parallelogram whose vertices are A (2,-1), B (0,2), C (2,3) and D (4,0) is
Options
- A⁻¹ 2
- B⁻¹ ( 1 3 )
- C⁻¹ 2
- D⁻¹ ( 2 3 )
Correct answer
C. ⁻¹ 2
Step-by-step solution
The acute angle between the diagonals can be found using slope analysis or vector methods. Given vertices A(2,-1) , B(0,2) , C(2,3) , and D(4,0) , diagonal AC is vertical with undefined slope, while diagonal BD has slope m = 0-2 4-0 = - 1 2 . For a vertical line and another with slope m , the acute angle satisfies = | 1 m | = 2 , giving = ⁻¹(2) . Alternatively, vectors AC = (0,4) and BD = (4,-2) yield dot product -8 and magnitudes 4 and 2 5 . The cosine of the full angle is - 1 5 , so the acute angle satisfies = 1