Let P(x,y)P(x,y) be any point on the /_-"bisector"∠−bisector of /_A∠A.
Then, from Geometry, we know that, PP is equidistant
from the sides AB and ACABandAC.
A(-1,1), & B(-9,-8) rArr the eqn. of A(−1,1),&B(−9,−8)⇒theeqn.ofAB# is given by,
AB : y-1={(1-(-8))/(-1-(-9))}(x-(-1)), i.e., AB:y−1={1−(−8)−1−(−9)}(x−(−1)),i.e.,
AB : 8(y-1)=9(x+1), or, 9x-8y+17=0AB:8(y−1)=9(x+1),or,9x−8y+17=0.
Similarly, AC : 3x+16y-13=0AC:3x+16y−13=0.
Now, the bot-"distance "d_1 ⊥−distance d1 from P" to "ABP to AB is given by,
d_1=|9x-8y+17|/sqrt(9^2+(-8)^2)=|9x-8y+17|/sqrt145d1=|9x−8y+17|√92+(−8)2=|9x−8y+17|√145.
The bot-"distance "d_2 ⊥−distance d2 from P" to "ACP to AC is,
d_2=|3x+16y-13|/sqrt(3^2+16^2)=|3x+16y-13|/sqrt265d2=|3x+16y−13|√32+162=|3x+16y−13|√265.
But, d_1=d_2d1=d2.
:. |9x-8y+17|/sqrt145=|3x+16y-13|/sqrt265.
:. (9x-8y+17)/sqrt29=+-(3x+16y-13)/sqrt53, are the desired
eqns.