# Worksheet on Rectangular – Polar Conversion

In math worksheet on rectangular – polar conversion; students can practice the questions on how to convert rectangular coordinates to polar coordinates and also convert polar coordinates to rectangular coordinates (vice-versa).

### Recall the formula from polar to rectangular:

To convert polar coordinates to rectangular coordinates;

x = r cos θ, y = r sin θ

### Recall the formula from rectangular to polar:

To convert rectangular coordinates to polar coordinates;

r = √(x² + y²) and tan θ = y/x or, θ = tan$$^{-1}$$ y/x

Follow the above formula to solve the below questions given in the worksheet on rectangular – polar conversion.

1. OX and OY are the cartesian axes of co-ordinates. Again 0 and OX are respectively the pole and initial line of a system of polar co-ordinates. With respect to these systems (i) if the polar co-ordinates of a point P be (2, 300), find the cartesian co-ordinates of the point; (ii) if the cartesian co-ordinates of a point P be (0, 2), find its polar co-ordinates.

2. Find the Cartesian co-ordinates of the points whose polar co-ordinates are :

(i) (2, π/3)

(ii) (4, 3π/2)

(iii) (6, -π/6)

(iv) (-4, π/3)

(v) (1, √3).

3. Find the polar co-ordinates of the points whose Cartesian co-ordinates are:

(i) (2, 2).

(ii) (- √3, 1)

(iii) (- 1, 1)

(iv) (1, - 1)

(v) (-(5√3)/2, - 5/2).

4. Reduce each of the following Cartesian equations to polar forms:

(i) x² + y² = a²

(ii) y = x tan α

(iii) x cos α + y sin α = p

(iv) y² = 4x + 3

(v) x² - y² = a²

(vi) x² + y² = 2ax

(vii) (x² + y²)² = a²(x² - y²)

5. Transform each of the following polar equations to cartesian forms:

(i) r = 2a sin θ

(ii) l/r = A cos θ + B sin θ

(iii) r= a sin θ

(iv) r² = a²cos 2θ

(v) $$r^{\frac{1}{2}}$$ $$a^{\frac{1}{2}}$$ sin θ/2

(vi) r² sin 2θ = 2a²

(vii) r cos (θ - α)

(viii) r(cos 3θ + sin 3θ) = 5k sin θ cos θ.

Answers for the worksheet on rectangular – polar conversion are given below to check the exact answers of the above questions.

1. (i) (√3 ,1)

(ii) (2, π/2);

2. (i) (1, √3)

(ii) (0, -4)

(iii) (3√3, -3)

(iv) (-2, -2√3),

(v) (cos √3, sin √3) where √3 is measured in radian.

3.(i) (2√2, π/4)

(ii) (2, 5π/6)

(iii) (√2, 3π/4)

(iv) (√2, -π/4)

(v) (5, 7π/6)

4. (i) r² = a²

(ii) θ = α

(iii) r cos (θ - α) = P

(iv) r² sin² θ = 4r cos θ + 3

(v) r² cos 2θ = a²

(vi) r = 2a cos θ

(vii) r² = a² cos 2θ.

5. (i) x² + y² = 2ay

(ii) Ax + By = l

(iii) x² + y² = ay

(iv) (x² + y²)² = a²(x² - y²)

(v) (2x² + 2y² + ax)² = a²(x² + y²)

(vi) xy = a²

(vii) x cos α + y sin α = p

(viii) x³ + 3x²y - 3xy² - y³ = 5kxy.

Co-ordinate Geometry

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