Video Transcript
Convert the polar equation 𝑟
equals four cos 𝜃 minus six sin 𝜃 to the rectangular form.
Remember, we convert from polar
coordinates to Cartesian coordinates or rectangular coordinates using the following
formulae. 𝑥 is equal to 𝑟 cos 𝜃 and 𝑦 is
equal to 𝑟 sin 𝜃. And these are suitable for all
values of 𝑟 and 𝜃. Our aim is going to be to
manipulate each of our equations so that we have an equation for cos 𝜃 and sin
𝜃.
Well, if we divide both sides of
our first equation by 𝑟, we see that cos 𝜃 is equal to 𝑥 over 𝑟. Similarly, dividing through by 𝑟
in our second equation, and we find sin 𝜃 equals 𝑦 over 𝑟. We can then replace cos 𝜃 with 𝑥
over 𝑟 and sin 𝜃 with 𝑦 over 𝑟 in our original polar equation. And we see that 𝑟 is equal to four
times 𝑥 over 𝑟 minus six times 𝑦 over 𝑟. This simplifies to four 𝑥 over 𝑟
minus six 𝑦 over 𝑟.
We’re next going to multiply
everything through by 𝑟. And we find that 𝑟 squared equals
four 𝑥 minus six 𝑦. Now we’re obviously not quite
done. We want to convert to rectangular
form. This is usually of the form 𝑦 is
equal to some function of 𝑥, although we’re essentially looking for an equation
with 𝑥 and 𝑦 as its only variables. So we recall the other conversion
formulae that we use to convert Cartesian to polar coordinates. That’s 𝑟 squared equals 𝑥 squared
plus 𝑦 squared. We should be able to see now that
we can replace 𝑟 squared with 𝑥 squared plus 𝑦 squared. So 𝑥 squared plus 𝑦 squared
equals four 𝑥 minus six 𝑦.
We’re almost there. You might recognise this
equation. We’re going to manipulate it by
completing the square. We subtract four 𝑥 from both sides
and add six 𝑦. Then we’re going to complete the
square for 𝑥 and 𝑦. We halve the coefficient of 𝑥 —
that gives us negative two — and then subtract negative two squared. So we subtract four. Similarly, we halve the coefficient
of 𝑦 to get three and then subtract three squared, which is nine. And of course, this is all equal to
zero. Negative four minus nine is
negative 13. So we add 13 to both sides of our
equation. And in rectangular form, our
equation is 𝑥 minus two all squared plus 𝑦 plus three all squared equals 13.
