From an aeroplane vertically over a straight | vedclass

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(D) Let $h$ be the height of the aeroplane above the road.Let the two milestones be at points $A$ and $B$,and the point on the road directly below the

Vedclass · Accepted Answer

$\frac{	an \alpha \cdot 	an \beta}{	an \alpha + 	an \beta}$

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(D) Let $h$ be the height of the aeroplane above the road.Let the two milestones be at points $A$ and $B$,and the point on the road directly below the aeroplane be $P$.Given that the distance between the two consecutive milestones is $1 	ext{ mile}$,so $AP + PB = 1$.In the right-angled triangles formed,we have $AP = h \cot \alpha$ and $PB = h \cot \beta$.Thus,$h \cot \alpha + h \cot \beta = 1$.$h(\cot \alpha + \cot \beta) = 1$.$h = \frac{1}{\cot \alpha + \cot \beta}$.Since $\cot 	heta = \frac{1}{	an 	heta}$,we have $h = \frac{1}{\frac{1}{	an \alpha} + \frac{1}{	an \beta}} = \frac{1}{\frac{	an \beta + 	an \alpha}{	an \alpha \cdot 	an \beta}} = \frac{	an \alpha \cdot 	an \beta}{	an \alpha + 	an \beta}$.

From an aeroplane vertically over a straight horizontal road,the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be $\alpha$ and $\beta$. Then,the height in miles of the aeroplane above the road is:

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