Heights and Distances

Interactive Feed

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is 30°. After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is 60°. Then the time taken (in minutes) by him, from B to reach the pillar, is:

If the angles of elevation of the top of a tower from three collinear points $A, B$ and $C$ , on a line leading to the foot of the tower, are $3 0^{\circ}, 4 5^{\circ}$ and $6 0^{\circ}$ respectively, then the ratio, $A B : B C$ , is:

1 : 3

2 : 3

3 : 1

3 : 2

A bird is sitting on the top of a vertical pole $20 m$ high and its elevation from a point $O$ on the ground is $4 5^{\circ}$ . It flies off horizontally straight away from the point $O$ . After one second, the elevation of the bird from $O$ is reduced to $3 0^{\circ}$ . Then the speed (in m/s) of the bird is

202

20 (3 - 1)

40 (2 - 1)

40 (3 - 2)

A particle just clears a wall of height b at a distance a and strikes the ground at a distance c from the point of projection. The angle of projection is

tan^{- 1} \frac{b c}{a ( c - a )}

tan^{- 1} \frac{b c}{a}

tan^{- 1} \frac{b}{a c}

A tower stands at the centre of a circular park. $A$ and $B$ are two points on the boundary of the park such that $A B (= a)$ subtends an angle of $6 0^{\circ}$ at the foot of the tower, and the angle of elevation of the top of the tower from $A$ or $B$ is $3 0^{\circ}$ . The height of the tower is

a / 3

a 3

2 a / 3

2 a 3

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $6 0^{\circ}$ and when he retires 40 meters away from the tree the angle of elevation becomes $3 0^{\circ}$ . The breadth of the river is

60 m

30 m

40 m

20 m

The upper $\frac{3}{4}$ th portion of a vertical pole subtends an angle $tan^{- 1} \frac{3}{5}$ at a point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is

JEE Advanced 2001

A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of $3 0^{\circ}$ . After some time, the angle of depression becomes $6 0^{\circ}$ . The distance (in metres) travelled by the car during this time is

1003

2003 /3

1003 /3

2003

JEE Advanced 2000

A pole stands vertically inside a triangular park $Δ A B C$ . If the angle of elevation of the top of the pole from each corner of the park is same, then in $Δ A B C$ the foot of the pole is at the

JEE Advanced 1983

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is $1 5^{\circ}$ . The distance of the boat from the foot of the light house is

(\frac{3 - 1}{3 + 1}) 60

(\frac{3 + 1}{3 - 1}) 60

(\frac{3 + 1}{3 - 1})^{2}

JEE Advanced 1998

LEVELJEE Advanced

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose $6 0^{\circ}$ and $3 0^{\circ}$ are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is at the points $P$ and $Q$ respectively on its path. Let $θ$ be the angle of elevation of the bird when it is a point on the arc of the circle exactly midway between $P$ and $Q$ . Find the numerical value of $tan^{2} θ$ . (Assume that the observer is not inside the vertical projection of the path of the bird.)

JEE Advanced 1993

LEVELJEE Advanced

An observer at $O$ notices that the angle of elevation of the top of a tower is $3 0^{\circ}$ . The line joining $O$ to the base of the tower makes an angle of $tan^{- 1} (1/ 2)$ with the North and is inclined Eastwards. The observer travels a distance of $300$ meters towards the North to a point $A$ and finds the tower to his East. The angle of elevation of the top of the tower at $A$ is $ϕ$ . Find $ϕ$ and the height of the tower.

JEE Advanced 1990

LEVELJEE Advanced

A vertical tower $P Q$ stands at a point $P$ . Points $A$ and $B$ are located to the South and East of $P$ respectively. $M$ is the mid point of $A B$ . $P A M$ is an equilateral triangle; and $N$ is the foot of the perpendicular from $P$ on $A B$ . Let $A N = 20$ metres and the angle of elevation of the top of the tower at $N$ is $tan^{- 1} (2)$ . Determine the height of the tower and the angles of elevation of the top of the tower at $A$ and $B$ .

JEE Advanced 1989

$A B C$ is a triangular park with $A B = A C = 100$ m. A television tower stands at the midpoint of $B C$ . The angles of elevation of the top of the tower at $A, B, C$ are $4 5^{\circ}, 6 0^{\circ}, 6 0^{\circ}$ , respectively. Find the height of the tower.

JEE Advanced 1988

A sign-post in the form of an isosceles triangle $A B C$ is mounted on a pole of height $h$ fixed to the ground. The base $B C$ of the triangle is parallel to the ground. A man standing on the ground at a distance $d$ from the sign-post finds that the top vertex $A$ of the triangle subtends an angle $β$ and either of the other two vertices subtends the same angle $α$ at his feet. Find the area of the triangle.

JEE Advanced 1983

Four ships $A, B, C$ and $D$ are at sea in the following relative positions : $B$ is on the straight line segment $A C$ , $B$ is due North of $D$ and $D$ is due west of $C$ . The distance between $B$ and $D$ is $2$ km. $∠ B D A = 4 0^{\circ}, ∠ B C D = 2 5^{\circ}$ . What is the distance between $A$ and $D$ ? [Take $sin 2 5^{\circ} = 0.423$ ]

JEE Advanced 1982

A vertical pole stands at a point $Q$ on a horizontal ground. $A$ and $B$ are points on the ground, $d$ meters apart. The pole subtends angles $α$ and $β$ at $A$ and $B$ respectively. $A B$ subtends an angle $γ$ at $Q$ . Find the height of the pole.

JEE Advanced 1994

A tower $A B$ leans towards west making an angle $α$ with the vertical. The angular elevation of $B$ , the topmost point of the tower is $β$ as observed from a point $C$ due west of $A$ at a distance $d$ from $A$ . If the angular elevation of $B$ from a point $D$ due east of $C$ at a distance $2 d$ from $C$ is $γ$ , then prove that $2 tan α = cot β + cot γ$ .

JEE Advanced 1991

A man notices two objects in a straight line due west. After walking a distance $c$ due north he observes that the objects subtend an angle $α$ at his eye; and, after walking a further distance $2 c$ due north, an angle $β$ . Show that the distance between the objects is $\frac{8 c}{3 c o t β - c o t α}$ ; the height of the man is being ignored.

JEE Advanced 1985

A ladder rests against a wall at an angle $α$ to the horizontal. Its foot is pulled away from the wall through a distance $a$ , so that it slides a distance $b$ down the wall making an angle $β$ with the horizontal. Show that $a = b tan \frac{1}{2} (α + β)$ .

JEE Advanced 1980

LEVELJEE Advanced

(i) $P Q$ is a vertical tower. $P$ is the foot and $Q$ is the top of the tower. $A, B, C$ are three points in the horizontal plane through $P$ . The angles of elevation of $Q$ from $A, B, C$ are equal, and each is equal to $θ$ . The sides of the triangle $A B C$ are $a, b, c$ ; and the area of the triangle $A B C$ is $Δ$ . Show that the height of the tower is $\frac{ab c t a n θ}{4Δ}$ . (ii) $A B$ is a vertical pole. The end $A$ is on the level ground. $C$ is the middle point of $A B$ . $P$ is a point on the level ground. The portion $C B$ subtends an angle $β$ at $P$ . If $A P = n A B$ , then show that $tan β = \frac{n}{2 n ^{2} + 1}$ .

JEE Advanced 1979

LEVELJEE Advanced

(a) A balloon is observed simultaneously from three points $A, B$ and $C$ on a straight road directly beneath it. The angular elevation at $B$ is twice that at $A$ and the angular elevation at $C$ is thrice that at $A$ . If the distance between $A$ and $B$ is $a$ and the distance between $B$ and $C$ is $b$ , find the height of the balloon in terms of $a$ and $b$ . (b) Find the area of the smaller part of a disc of radius $10$ cm, cut off by a chord $A B$ which subtends an angle of $22 \frac{1}{2}^{\circ}$ at the circumference.

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that AP = 2AB. If $∠ B P C = β$ , then $tan β$ is equal to:

\frac{6}{7}

\frac{1}{4}

\frac{2}{9}

\frac{4}{9}

JEE Main 2018 (Paper 1)

$P QR$ is a triangular park with $P Q = P R = 200$ m. A T.V. tower stands at the mid-point of $QR$ . If the angles of elevation of the top of the tower at $P, Q$ and $R$ are respectively $4 5^{\circ}, 3 0^{\circ}$ and $3 0^{\circ}$ , then the height of the tower (in m) is :

JEE Main 2018 (15 April Shift 1)

An aeroplane flying at a constant speed, parallel to the horizontal ground, $3$ km above it, is observed at an elevation of $6 0^{\circ}$ from a point on the ground. If, after five seconds, its elevation from the same point, is $3 0^{\circ}$ , then the speed (in km/hr) of the aeroplane, is :