+14995 The coordinates given the circle are not correkt. (Thanks ElectricPavlov!)
1. Determination of the slope of the chords between the points AO and BA.
\(m_{AO}=\frac{y_A}{x_A}=\frac{1}{-2}=\color{blue}-0.5\\ m_{BA}=\frac{y_B-y_a}{x_B-x_A}=\frac{2-1}{-3-(-2)}=\color{blue}-1\\\)
2. Determination of the coordinates of the centers of the chords BA and AO.
\( P_{BA}(\frac{x_B+x_A}{2},\frac{y_B+y_a}{2})\\ P_{BA}(\frac{-3+(-2)}{2},\frac{2+1}{2})\\ \color{blue}P_{BA}(-2.5,1.5) \)
\(P_{AO}(\frac{x_A+x_O}{2},\frac{y_A+y_O}{2})\\ P_{AO}(\frac{-2+0}{2},\frac{1+0}{2})\\ \color{blue}P_{AO}(-1,0.5)\)
3. Point-direction functions for the center points of the chords BA and AO. The slopes of the function graphs are the negative reciprocals of the slopes of the chords.
\(f_{BA}(x)=-\frac{1}{m_{BA}}(x-x_{PBA})+y_{PBA}\\ f_{BA}(x)=-\frac{1}{-1}(x-(-2.5))+1.5\\ {\color{blue}f_{BA}(x)}=1(x+2.5)+1.5\color{blue}=x+4\\ f_{AO}(x)=-\frac{1}{m_{AO}}(x-x_{P_{AO}})+y_{P_{AO}}\\ f_{AO}(x)=-\frac{1}{-0,5}(x-(-1))+0.5\\ {\color{blue}f_{AO}(x)}=2(x+1)+0.5\color{blue}=2x+2.5 \)
4. Determine the coordinates of point P by equating the two functions.
\(x+4=2x+2.5\\ \color{blue}x_P=1.5\\ y_P=x_P+4=1.5+4\\ \color{blue}y_P=5.5\\ .\\ \color{blue}P(1.5,5.5)\)
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