Consider the coordinates of C as (a,b), and since the point C lies on the angular bisector of C, therefore the equation becomes,
7a−4b=1 (1)
The midpoint of AC will be (a−32,b+12) and as the median lies on the midpoint of B, therefore the equation becomes,
2(a−32)+(b+12)=3
2a−6+b+1=6
2a+b=11 (2)
From equation (1) and (2),
a=3,b=5
Slope of AC is,
m1=5−13−(−3)
=23
The slope of bisector is,
m2=−Coeff.ofxCoeff.ofy
=74
The angle between AC and the bisector will be,
tanα=∣∣∣m1−m21+m1m2∣∣∣
=∣∣ ∣ ∣ ∣∣23−741+(23)(74)∣∣ ∣ ∣ ∣∣
=12
Let m be the slope of BC.
The angle between BC and bisector will be equal to the angle between AC and bisector.
∣∣∣m−m21+mm2∣∣∣=12
∣∣ ∣ ∣∣m−741+m(74)∣∣ ∣ ∣∣=12
4m−77m+4=±12
m=23;m=18
When the slope of BC is 23, then A,B,C will be collinear and it will form no triangle.
Therefore, the slope of BC is 18.