Question

20. The circle \( \mathcal{C} \) in the plane centred at the point \( \left(x_{1}, y_{1}\right) \) with radius \( r \) satisfies the equation \[ \left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}=r^{2} \] Let \( \mathcal{C}^{\prime} \) be the circle that results by rotating \( \mathcal{C} \) ninety degrees anticlockwise about the point with coordinates \( \left(x_{0}, y_{0}\right) \). Find the equation describing \( \mathcal{C}^{\prime} \). \[ \left(x-x_{0}-y_{0}+y_{1}\right)^{2}+\left(y+y_{0}+x_{0}-x_{1}\right)^{2}=r^{2} \] \[ \left(x+x_{0}+y_{0}-y_{1}\right)^{2}+\left(y+y_{0}-x_{0}+x_{1}\right)^{2}=r^{2} \] \[ \left(x-x_{0}-y_{0}+y_{1}\right)^{2}+\left(y-y_{0}+x_{0}-x_{1}\right)^{2}=r^{2} \] \[ \left(x-x_{0}+y_{0}-y_{1}\right)^{2}+\left(y-y_{0}-x_{0}+x_{1}\right)^{2}=r^{2} \] \[ \left(x+x_{0}-y_{0}+y_{1}\right)^{2}+\left(y-y_{0}+x_{0}-x_{1}\right)^{2}=r^{2} \]

          20. The circle \( \mathcal{C} \) in the plane centred at the point \( \left(x_{1}, y_{1}\right) \) with radius \( r \) satisfies the equation
\[
\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}=r^{2}
\]

Let \( \mathcal{C}^{\prime} \) be the circle that results by rotating \( \mathcal{C} \) ninety degrees anticlockwise about the point with coordinates \( \left(x_{0}, y_{0}\right) \). Find the equation describing \( \mathcal{C}^{\prime} \).
\[
\left(x-x_{0}-y_{0}+y_{1}\right)^{2}+\left(y+y_{0}+x_{0}-x_{1}\right)^{2}=r^{2}
\]
\[
\left(x+x_{0}+y_{0}-y_{1}\right)^{2}+\left(y+y_{0}-x_{0}+x_{1}\right)^{2}=r^{2}
\]
\[
\left(x-x_{0}-y_{0}+y_{1}\right)^{2}+\left(y-y_{0}+x_{0}-x_{1}\right)^{2}=r^{2}
\]
\[
\left(x-x_{0}+y_{0}-y_{1}\right)^{2}+\left(y-y_{0}-x_{0}+x_{1}\right)^{2}=r^{2}
\]
\[
\left(x+x_{0}-y_{0}+y_{1}\right)^{2}+\left(y-y_{0}+x_{0}-x_{1}\right)^{2}=r^{2}
\]

20. The circle 𝒞 in the plane centred at the point (x1, y1) with radius r satisfies the equation

(x-x1)^2+(y-y1)^2=r^2

Let 𝒞^' be the circle that results by rotating 𝒞 ninety degrees anticlockwise about the point with coordinates (x0, y0). Find the equation describing 𝒞^'.

(x-x0-y0+y1)^2+(y+y0+x0-x1)^2=r^2

(x+x0+y0-y1)^2+(y+y0-x0+x1)^2=r^2

(x-x0-y0+y1)^2+(y-y0+x0-x1)^2=r^2

(x-x0+y0-y1)^2+(y-y0-x0+x1)^2=r^2

(x+x0-y0+y1)^2+(y-y0+x0-x1)^2=r^2

Added by Randy R.

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