Question

1. A jet plane is at a vertical height of h. The angles of depression of two tanks on the ground in the same line with the plane are alpha
and Beta (alpha>Beta).Prove that the distance between the tank is h(tan alpha-tanbeta)/(tan alpha .tanbeta)​

Answer 1

Let A be the position of the plane and C and D be the tanks as shown in the figure

Let the distance between the tanks be x

Then

In ΔABC

$\tan\alpha=\frac{AB}{BC}$

$\implies \tan\alpha=\frac{h}{BC}$

$\implies BC=\frac{h}{\tan\alpha}$

In In ΔABD

$\tan\beta=\frac{AB}{BD}$

$\implies \tan\beta=\frac{h}{BC+BD}$

$\implies \tan\beta=\frac{h}{h/\tan\alpha + x}$

$\implies \tan\beta(\frac{h}{\tan\alpha}+x)=h$

$\implies h\tan\beta+x\tan\alpha \tan\beta=h\tan\alpha$

$\implies x\tan\alpha \tan\beta=h(\tan\alpha-\tan\beta)$

$\implies x=\frac{h(\tan\alpha-\tan\beta)}{\tan\alpha \tan\beta}$

Hope this helps.