How do you identify the type of graph generated by the polar equation $r = 2 a sin θ + 2 a cos θ$ $r=2asintheta+2acostheta$ and what is the equivalent rectangular equation?

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Answer 1 · 2018-01-07T09:31:22

see below

convert to rectangular first

$r^{2} = x^{2} + y^{2} - - (1)$ $r^2=x^2+y^2--(1)$

$x = r cos θ - - (2)$ $x=rcostheta--(2)$

$y = r sin θ - - (3)$ $y=rsintheta--(3)$

$r = 2 a sin θ + 2 a cos θ - - (4)$ $r=2asintheta+2acostheta--(4)$

$(4) \times r$ $(4)xxr$

$r^{2} = 2 a r sin θ + 2 a r cos θ$ $r^2=2arsintheta+2arcostheta$

using $(1), (2), (3)$ $(1),(2),(3)$

$x^{2} + y^{2} = 2 a y + 2 a x$ $x^2+y^2=2ay+2ax$

$x^{2} - 2 a x + y^{2} - 2 a y = 0$ $x^2-2ax+y^2-2ay=0$

complete the square ( the steps are omitted as an exercise for the reader)

${(x - a)}^{2} + {(y - a)}^{2} - 2 a^{2} = 0$ $(x-a)^2+(y-a)^2-2a^2=0$

which is a circle centre $(a, a)$ $" "(a,a)$ radius $a \sqrt{2}$ $" "asqrt(2)$

How do you identify the type of graph generated by the polar equation r=2asintheta+2acosthetar=2asinθ+2acosθr=2asintheta+2acostheta and what is the equivalent rectangular equation?