Exams Joint Entrance Examination Mathematics Straight Lines and Pair of Straight Lines Question 12 Question 12 of 303 Medium MCQ
Let a point A lie between the parallel lines L 1 {L}_{1} L 1 and L 2 {L}_{2} L 2 such that its distances from L 1 {L}_{1} L 1 and L 2 {L}_{2} L 2 are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines L 1 {L}_{1} L 1 and L 2 {L}_{2} L 2 , respectively, is :
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a \triangleABC a\triangleABC a \triangleABC
sin θ = 3 a sin ( 60 ∘ + θ ) = 9 a ⇒ 3 2 × 1 − 9 a 2 + 3 a × 1 2 = 9 a \begin{gathered} \sin \theta =\frac{3}{a} \\ \sin ({60}^{\circ }+\theta )=\frac{9}{a} \\ \Rightarrow \frac{\sqrt{3}}{2}\times \sqrt{1-\frac{9}{{a}^{2}}}+\frac{3}{a}\times \frac{1}{2}=\frac{9}{a} \end{gathered} sin θ = a 3 sin ( 60 ∘ + θ ) = a 9 ⇒ 2 3 × 1 − a 2 9 + a 3 × 2 1 = a 9
⇒ 3 ( a 2 − 9 ) + 3 = 18 ⇒ 3 ( a 2 − 9 ) = 15 2 ⇒ a 2 − 9 = 15 × 5 ⇒ a 2 = 84 = 3 4 a 2 = 21 3 \begin{gathered} \Rightarrow \sqrt{3}(\sqrt{{a}^{2}-9})+3=18 \\ \Rightarrow 3({a}^{2}-9)={15}^{2}\, \Rightarrow {a}^{2}-9=15\times 5 \\ \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \Rightarrow {a}^{2}=84 \\ =\frac{\sqrt{3}}{4}{a}^{2}=21\sqrt{3} \end{gathered} ⇒ 3 ( a 2 − 9 ) + 3 = 18 ⇒ 3 ( a 2 − 9 ) = 15 2 ⇒ a 2 − 9 = 15 × 5 ⇒ a 2 = 84 = 4 3 a 2 = 21 3 Answer the question to unlock