Points, Vectors, and Functions Exercises

Answer

Translate each equation into polar coordinates.

• We can square the terms first or replace x and y first. If we square the terms first, we find

Rearranging, and remembering that we can replace x² + y² with r², we find

Since we have r² and r terms, it's not reasonable to isolate r by itself.

• Replace x and y to find

If r ≠ 0 then we can divide both sides by r to find

• Square first to find

Then replace:

It's reasonable to separate out r² in this equation.

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Answer

Replace y to find

Replace x and y to find

Replace x² + y² with r² to find

Therefore

r < 10.

Answer

• Plug in x and y wherever possible.

  r cos θ + r² sin² θ = 4r sin θ

  r cos θ + (r sin θ)² = 4r sin θ

  x + y²  = 4y.

• Multiply both sides by r:

  r sin θ = 5r cos θ + 2

  y = 5x + 2

• Square both sides and then substitute:

  r = 4

  r² = 16

  x² + y² = 16

Answer

1.  Multiply both sides by r to find

   

2. Factor out r on the left-hand side:

   r cos² θ + r sin² θ =  1

   r(cos² θ + sin² θ) =  1

   r  =  1

   Then square both sides:

   r  = 1

   r² = 1

   x² + y² =  1

3. Substitute in x and y first:

   5r² cos² θ + 10r cos θ + 5 – 4r²sin² θ + 16r sin θ – 16  =  20

   5(r cos θ)² + 10r cos θ + 5 – 4(r sin θ)² + 16r sin θ – 16  =  20

   5x² + 10x + 5 – 4y² + 16y – 16  =  20.

   Now we can factor to tidy things.

   5x² + 10x + 5 – 4y² + 16y – 16 = 20

   5(x² + 2x + 1) – 4(y²– 4y + 4) = 20

   5(x + 1)² – 4(y – 2)² = 20.

   Dividing both sides by 20, we see that this is the equation of a hyperbola:

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