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JEE Main 2019
Height and Distance
Height and Distance
Hard

Question

From the top A\mathrm{A} of a vertical wall AB\mathrm{AB} of height 30 m30 \mathrm{~m}, the angles of depression of the top P\mathrm{P} and bottom Q\mathrm{Q} of a vertical tower PQ\mathrm{PQ} are 1515^{\circ} and 6060^{\circ} respectively, B\mathrm{B} and Q\mathrm{Q} are on the same horizontal level. If C\mathrm{C} is a point on AB\mathrm{AB} such that CB=PQ\mathrm{CB}=\mathrm{PQ}, then the area (in m2\mathrm{m}^{2} ) of the quadrilateral BCPQ\mathrm{BCPQ} is equal to :

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Solution

Given, ABA B be a vertical wall of height 30 m30 \mathrm{~m} and PQP Q be a vertical tower. Such that BQA=TAQ=60\angle B Q A=\angle T A Q=60^{\circ} (Alternate angles) and CPA=TAP=15\angle C P A=\angle T A P=15^{\circ} Now, tan60=ABBQ=30BQ\tan 60^{\circ}=\frac{A B}{B Q}=\frac{30}{B Q} 3=30BQBQ=303=103\begin{array}{ll} &\Rightarrow \sqrt{3}=\frac{30}{B Q} \\\\ &\Rightarrow B Q=\frac{30}{\sqrt{3}}=10 \sqrt{3} \end{array}  and tan15=ACPC23=AC103(PC=BQ)AC=103(23)=20330BC=ABAC=30(20330)=60203\begin{aligned} & \text { and } \tan 15^{\circ}=\frac{A C}{P C} \\\\ & \Rightarrow 2-\sqrt{3}=\frac{A C}{10 \sqrt{3}} (\because P C=B Q) \\\\ & \Rightarrow A C=10 \sqrt{3}(2-\sqrt{3})=20 \sqrt{3}-30 \\\\ & \therefore B C=A B-A C=30-(20 \sqrt{3}-30)=60-20 \sqrt{3} \end{aligned} \therefore Area of quadrilateral BCPQB C P Q =BQ×BC=103×(60203)=10(60360)=600(31)m2\begin{aligned} & =B Q \times B C=10 \sqrt{3} \times(60-20 \sqrt{3}) \\\\ & =10(60 \sqrt{3}-60) \\\\ & =600(\sqrt{3}-1) \mathrm{m}^2 \end{aligned}

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