Question

Question #87e23

Answer 1

`x^2+y^2+4y-8x=5`

Rewritten as the standard form of a circle:
`(x-4)^2+(y+2)^2=25`

Explanation:

`r^2+4rsintheta-8rcostheta=5`

Formulas to remember:
`color(blue)(y=rsintheta)`
`color(blue)(x=rcostheta)`
`color(blue)(x^2+y^2=r^2)`

Substituting these rectangular variables into the polar form, we get:
`x^2+y^2+4y-8x=5`

To get this into standard form of a conic section, we have to complete the square for both `x` and `y`.

Subtract `5` from each side (optional):
`x^2-8x+y^2+4y-5=0`

Complete the square for each variable (taking into account the constant being added):
`(x-4)^2color(red)(-16)+(y+2)^2color(red)(-9)=0`

Add `25` to each side:
`(x-4)^2+(y+2)^2=color(red)(25)`