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| To find the equation of the bisectors of the angle between ax+by+c=0 and ax1+by1+c1=0 |
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| To find the equation of the bisectors of the angle between the lines ax+by+c=0 and ax1+by1+c1=0 |
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| Let the lines AB and CD intersect at S. |
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| Let P(x,y) be a point on the angle bisector of any one of the angles. Let P(x,y) be on SU, then the length PM of the perpendicular drawn from P on AB = PL, the length drawn to CD. |
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| The origin and the point P are on the same side of both the lines SA and SD. |
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| \ ax + by + c
and a1x + b1y + c1 will both be positive as
C and C1 are both positive. |
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| Now the above ratio is true for all points on SU. |
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| The equation of the angle bisector SU is |
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 . … (i) |
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| Here O and P are on the opposite sides of SD. Therefore C and |
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| ax + by + c must be of opposite signs. But C > 0, ax + by + c < 0 |
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| or -(ax + by + c) > 0. Again, since O and P are on the same side of the line |
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| The point P(x,y) lies on the bisector ST. |
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| \The length of PQ, the perpendicular drawn from P on SD = perpendicular PR drawn from P onto SB. |
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| The relation holds good for all points on the bisector ST, therefore the equation of the bisectors ST is |
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| Using (i) and (ii), the equation of the bisectors of the angles can be written as |
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