Olympiad 3

A
B
C
Exercise 32. In the coordinate plane, given points A(0, 1.5) and B(3, 0).
1. Find the equation of the straight line joining the points A and B.
2. Express sin θ + 2 cos θ in the form r sin(θ + α), where r is a positive
number and α is an acute angle.
3. The figure shows a map of a moorland.
The units of the coordinates
are kilometers, and y−axis points
due north. A walker leaves her car
somewhere on the straight road between
A and B. She walks in the
straight line for distance 2 km to a
monumnet at the origin O.
y
(0, 1.5)
A
θ 2
Q
(3, 0)
B
x
While she is looking at the fog comes down, so that she cannot see the
way back to her car. She needs to work out the bearing on the which
she should walk.
Write down the coordinate of the point Q which is 2 km from O on a
bearing of θ. Show that, for Q to be on the road between A and B, θ
must satisfy the equation
2 sin θ + 4 cos θ = 3
Calculate θ between 0 and π 2
which satisfies this equation.
8 Score: