Q:-from the top of a tree on the bank of a lake , an aeroplane in the sky makes an angle of elevation or and its image in the river makes an angle of depression .if the height of the tree from the water surface is 'a' and the height of the aeroplane is 'h'. show that h=asin(a+b)/sin(b-a)

let B be the aero plane

D be the image
AC be y
BC be x
BC = h-x
CD = a + h
Consider ABC
Sin alpha = h-a / y

Y=h-a/ Sin alpha ------------ (1)


Consider ACD


Sin beta = h + a / y
Y= Sin beta / h + a ------------ (2)
From (1) & (2)
h-a/ Sin alpha = Sin beta / h + a

h-a (Sin beta) = h + a (Sin alpha)
h Sin beta-a Sin beta = h Sin alpha +a Sin alpha
h (Sin beta- Sin alpha) = a (Sin alpha - Sin beta)

h=a sin (alpha + beta) / sin (beta - alpha)
hence proved