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Revision difference : chordal |
| Version 4 |
Version 3 |
| By the \PMlinkescapetext{parent} entry, the power of the point $(a,\,b)$ with respect of the circle |
By the \PMlinkescapetext{parent} entry, the power of the point $(a,\,b)$ with respect of the circle |
| $$K_1(x,\,y) := (x-x_1)^2+(y-y_1)^2-r_1^2 =0$$ |
$$K_1(x,\,y) := (x-x_1)^2+(y-y_1)^2-r_1^2 =0$$ |
| is equal to \,$K_1(a,\,b)$\, and with respect to the circle |
is equal to \,$K_1(a,\,b)$\, and with respect to the circle |
| $$K_2(x,\,y) := (x-x_2)^2+(y-y_2)^2-r_2^2 =0$$ |
$$K_2(x,\,y) := (x-x_2)^2+(y-y_2)^2-r_2^2 =0$$ |
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equal to \,$K_2(a,\,b)$. \,Thus the locus of all points $(x,\,y)$ having the same \PMlinkescapetext{power} with respect to both circles is characterized by the equation
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equal to $K_2(a,\,b)$. \,Thus the locus of all points $(x,\,y)$ having the same \PMlinkescapetext{power} with respect to both circles is characterized by the equation
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| $$K_1(x,\,y) = K_2(x,\,y).$$ |
$$K_1(x,\,y) = K_2(x,\,y).$$ |
| This equation reduces to the form |
This equation reduces to the form |
| $$2(x_2-x_2)x+2(y_2-y_1)y+k = 0,$$ |
$$2(x_2-x_2)x+2(y_2-y_1)y+k = 0,$$ |
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and hence the locus is a straight line perpendicular to the \PMlinkescapetext{{\em centre line}} of the circles. \,This locus is called the {\em chordal} or the {\em radical axis} of the circles.
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and hence the locus is a straight line perpendicular to the {\em centre line} of the circles. This locus is called the {\em chordal} or the {\em radical axis} of the circles.
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