Suppose that \( P = (x, y) \) has polar coordinates \(\left( r, \frac{\pi}{15} \right)\). Find the polar coordinates for the following points if \( \theta \in [0, 2\pi] \).(a) \( P = (x, -y) \)(Give your answer in the form \((*, *)\). Express numbers in exact form. Use symbolic notation and fractions where needed.)**Polar coordinates:** \[ \left( \sqrt{x^2 + y^2}, 0 \right) \]*Incorrect*(b) \( P = (-x, -y) \)(Give your answer in the form \((*, *)\). Express numbers in exact form. Use symbolic notation and fractions where needed.)**Polar coordinates:**\[ \left( \sqrt{x^2 + y^2}, \pi \right) \]*Incorrect*(c) \( P = (-x, y) \)(Give your answer in the form \((*, *)\). Express numbers in exact form. Use symbolic notation and fractions where needed.)**Polar coordinates:**\[ \left( \sqrt{x^2 + y^2}, 0 \right) \]*Incorrect*(d) \( P = (y, x) \)(Give your answer in the form \((*, *)\). Express numbers in exact form. Use symbolic notation and fractions where needed.)**Polar coordinates:**\[ \left( \sqrt{x^2 + y^2}, 0 \right) \]*Incorrect*

Please see the answer below.P=(x,y) has polar coordinates r,π15.So, r=x2+y2, and π15=tan-1yx(a) P(x,-y)P is in Quadrant IVP=(x,-y) has polar coordinates (x)2+(-y)2, tan-1(-y)(x) =x2+y2, tan-1-yx=x2+y2, -tan-1yx+2π ......... As tan-1-x=- tan-1x=- tan-1x+2π, θ∈[0,2π], and P is in Quadrant IV=r,-π15+2π=r,29π15Answer: polar coordinates :r,29π15(b) P(-x,-y)P is in Quadrant IIIP=(-x,-y) has polar coordinates (-x)2+(-y)2, tan-1(-y)(-x) =x2+y2, tan-1yx+π ........ As tan-1x=tan-1x+π, θ∈[0,2π], and P is in Quadrant III=r,π15+π=r,16π15Answer: polar coordinates :r,16π15(c) P(-x,y)P=(-x,y) has polar coordinates (-x)2+(y)2, tan-1(y)(-x) =x2+y2, tan-1-yx=x2+y2, -tan-1yx+π ......... As tan-1-x=- tan-1x=- tan-1x+π, θ∈[0,2π], and P is in Quadrant II=r,-π15+π=r,14π15Answer: polar coordinates :r,14π15(d) P(y,x) P is in Quadrant IP=(y,x) has polar coordinates (y)2+(x)2, tan-1(x)(y) =x2+y2, tan-11yx=x2+y2, π2-tan-1yx ......... As tan-1x+tan-11x=π2, θ∈[0,2π], and P is in Quadrant I=r,π2-π15=r,13π30Answer: polar coordinates :r,13π30