# Two points on the polar plane are given as P(−2, 3π/2 ) and Q(2, 2π/3). Find the triangular area formed by P, Q, and the pole.

(A) 1/2 (B) 1 (C) √2 (D) √3

## (-2,3π/2) = (2, π/2)

angle between them is 2π/3 - π/2 = π/6 ....... (think of it as 120° - 90° = 30°)

area = (1/2)(2)(sin π/6 )
= (1/2)(4)(1/2) = 1 square unit

## To find the triangular area formed by P, Q, and the pole, we can use the formula for the area of a triangle:

Area = 1/2 * base * height

First, let's find the length of the base of the triangle. The base of the triangle is the distance between the two given points P and Q.

Using the distance formula in polar coordinates:

d = √((r2)^2 + (θ2 - θ1)^2)

where r1, r2 are the magnitudes (distances from the origin) of the two points, and θ1, θ2 are the angles of the two points.

For point P:
r1 = -2
θ1 = 3π/2

For point Q:
r2 = 2
θ2 = 2π/3

Calculating the distance (base) between P and Q:

d = √((2)^2 + (2π/3 - 3π/2)^2)
= √(4 + (4π/3 - 9π/6)^2)
= √(4 + (4π/3 - 3π/3)^2)
= √(4 + (π/3)^2)
= √(4 + π^2/9)
= √((36 + π^2)/9)
= (√(36 + π^2))/3

Now, let's find the height of the triangle. The height of the triangle is the distance between the pole and the line segment connecting P and Q.

Since the pole is at the origin (0,0), the distance from the origin to the line segment connecting P and Q is equal to the radius of either point.

Using the given points:
r1 = -2 (magnitude of point P)
r2 = 2 (magnitude of point Q)

Either r1 or r2 will give us the height of the triangle. Let's choose r1:

height = r1 = -2

Finally, let's calculate the triangular area using the formula:

Area = 1/2 * base * height
= 1/2 * (√(36 + π^2))/3 * (-2)
= -1/3 * (√(36 + π^2))

So, the triangular area formed by P, Q, and the pole is -1/3 * (√(36 + π^2)).

None of the answer choices match this result.

## To find the area of the triangle formed by the points P, Q, and the pole, we need to convert the given polar coordinates to Cartesian coordinates and then use the formula for the area of a triangle.

Let's convert the points P and Q from polar coordinates to Cartesian coordinates:

P(−2, 3π/2):
To convert polar coordinates to Cartesian coordinates, we use the formulas:
x = r * cos(θ)
y = r * sin(θ)

For P(−2, 3π/2):
r = -2
θ = 3π/2

x = -2 * cos(3π/2) = 0
y = -2 * sin(3π/2) = -2

So, the Cartesian coordinates of point P are (0, -2).

Q(2, 2π/3):
Using the same formulas for converting polar coordinates to Cartesian coordinates:

r = 2
θ = 2π/3

x = 2 * cos(2π/3) = -1
y = 2 * sin(2π/3) = √3

So, the Cartesian coordinates of point Q are (-1, √3).

Now, we have the Cartesian coordinates of points P and Q. Let's find the distance between these two points, which will be the base of the triangle formed by P, Q, and the pole.

Using the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)

distance = √((0 - (-1))^2 + (-2 - √3)^2)
distance = √((1)^2 + (-2 - √3)^2)
distance = √(1 + 4 + √12 - 4√3)
distance = √(5 + √12 - 4√3)

Now, let's calculate the height of the triangle. The height is the y-coordinate of the pole, which is 0.

The area of a triangle is given by the formula: area = (base * height) / 2

Substituting the values:
area = ((√(5 + √12 - 4√3)) * 0) / 2
area = 0

Therefore, the area of the triangular region formed by P, Q, and the pole is 0.