Exams Joint Entrance Examination Mathematics Straight Lines and Pair of Straight Lines Question 13 Question 13 of 303 Medium MCQ
Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle \[\alpha \] with the positive x-axis and the equations of its diagonals are ( 3 + 1 ) x + ( 3 − 1 ) y = 0 (\sqrt{3}+1)x+(\sqrt{3}-1)y=0 ( 3 + 1 ) x + ( 3 − 1 ) y = 0
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O B = 3 + 1 1 − 3 = tan 105 ∘ ∴ α = 60 ∘ ∴ A ( a c o s 60 ∘ , a s i n 60 ∘ ) ∴ A ( a 2 , 3 a 2 ) ∴ ( 3 − 1 2 ) a − ( 3 + 1 2 ) ⋅ 3 a + 8 3 = 0 a [ 3 − 1 − 3 − 3 2 ] = − 8 3 \begin{gathered} OB=\frac{\sqrt{3}+1}{1-\sqrt{3}} \\ =\tan {105}^{\circ } \\ \therefore \alpha ={60}^{\circ } \\ \therefore A(acos{60}^{\circ },asin{60}^{\circ }) \\ \therefore A(\frac{a}{2},\frac{\sqrt{3}a}{2}) \\ \\ \therefore (\frac{\sqrt{3}-1}{2})a-(\frac{\sqrt{3}+1}{2})\cdot \sqrt{3}a+8\sqrt{3}=0 \\ a[\frac{\sqrt{3}-1-3-\sqrt{3}}{2}]=-8\sqrt{3} \end{gathered} O B = 1 − 3 3 + 1 = tan 105 ∘ ∴ α = 60 ∘ ∴ A ( a cos 60 ∘ , a s in 60 ∘ ) ∴ A ( 2 a , 2 3 a ) ∴ ( 2 3 − 1 ) a − ( 2 3 + 1 ) ⋅ 3 a + 8 3 = 0 a [ 2 3 − 1 − 3 − 3 ] = − 8 3
a = 4 3 ∴ a 2 = 48 \begin{gathered} a=4\sqrt{3} \\ \therefore {a}^{2}=48 \end{gathered} a = 4 3 ∴ a 2 = 48 Answer the question to unlock