# Clinky observes a towwr PQ of height h from a point A on the ground . She moves a distance d towards the tower and find that the angle of elevation has direction and finals that the angle of elevation is 3 times at A . Prove that 36h2=35d2

1
by sjansanibint6an

2016-02-16T21:27:33+05:30
Let RB = x
BQR is ext of ΔPBQ
∴ PBQ = 2
Now inΔ PBQ, PBQ = QPB
PQ = QB = d
Also, BRA is ext of Δ BQR
∴ QBR = 3
And BRQ = 3 (Linear Pair)
Now in ΔBQR, by applying Sine Law, we get
d/sin (-3) = 3d/4 /sin = x/sin²
d/sin 3 = 3d/4 /sin = x/sin²
d/3 sin 4 sin³ = 3d/4 sin  = x/2sin  cos
d/3 4sin² = 3d/4 = x/2cos ..................(I)(II)(III)
From eq. (I),I=II
d/3 4sin² = 3d/4 ⇒4 = 9 12 sin²
sin² = 5/12 ⇒cos² = 7/12
Also from eq. (1) using (II) and (III) we have
3d/4 = x/2 cos⇒4x²² = 9 d²cos²
x² = 9d²/4 = 7/12 = 21/16 d²...............(3)
Again from ΔABR, we have sin 3 = h/x
3 sin  4 sin³ = h/x ⇒sin(3 4sin²) = h/x
sin(3 4 x5/9) = h/x (using sin² = 5/12)
4/3 sin = h/x
Squaring both sides, we get
16/9 sin² = h²/x²16/9x2/12 = h²/x²
(again using sin² = 5/12)
h² = 4 x 5/9 x 3 x²⇒ h² = 20/27 x 21/16 d²
{using value of x² from eq.(3)}⇒h² = 35/36 d²⇒36 h² = 35d²
Proved
Thank You.........