Question:
Let ABC be a triangle such that the vertex A has coordinates \((-3,1)\). If the equation of the median through B be \(2x+y-3=0\) and the equation of the bisector of angle C be \(7x-4y-1=0\), then the equation of the line BC is :
Let ABC be a triangle such that the vertex A has coordinates \((-3,1)\). If the equation of the median through B be \(2x+y-3=0\) and the equation of the bisector of angle C be \(7x-4y-1=0\), then the equation of the line BC is :
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Find C using the midpoint of AC on the median, then use the reflection of A in the angle bisector of C, which lies on BC.
Updated On: Nov 12, 2026
- \(15x+y=50\)
- \(4x-y=7\)
- \(18x-y=49\)
- \(7x-3y=6\)
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The Correct Option is C
Solution and Explanation
C lies on \(7x-4y-1=0\): let \(C=\left(c,\frac{7c-1}{4}\right)\).
The median through B passes through the midpoint of AC, which is \(\left(\frac{c-3}{2},\frac{7c+3}{8}\right)\). Substituting in \(2x+y-3=0\): \((c-3)+\frac{7c+3}{8}-3=0\Rightarrow 15c=45\Rightarrow c=3\). So \(C=(3,5)\).
The reflection of A in the bisector of angle C lies on line BC. For \(7x-4y-1=0\): value at A \(=-21-4-1=-26\), and \(7^{2}+4^{2}=65\). The reflection is \(A'=(-3,1)-2\cdot\frac{-26}{65}(7,-4)=\left(-3+\frac{28}{5},\,1-\frac{16}{5}\right)=\left(\frac{13}{5},-\frac{11}{5}\right)\).
Slope of \(CA'=\frac{5+\frac{11}{5}}{3-\frac{13}{5}}=\frac{36/5}{2/5}=18\).
Line BC: \(y-5=18(x-3)\Rightarrow 18x-y=49\).
Answer: (3) \(18x-y=49\)
The median through B passes through the midpoint of AC, which is \(\left(\frac{c-3}{2},\frac{7c+3}{8}\right)\). Substituting in \(2x+y-3=0\): \((c-3)+\frac{7c+3}{8}-3=0\Rightarrow 15c=45\Rightarrow c=3\). So \(C=(3,5)\).
The reflection of A in the bisector of angle C lies on line BC. For \(7x-4y-1=0\): value at A \(=-21-4-1=-26\), and \(7^{2}+4^{2}=65\). The reflection is \(A'=(-3,1)-2\cdot\frac{-26}{65}(7,-4)=\left(-3+\frac{28}{5},\,1-\frac{16}{5}\right)=\left(\frac{13}{5},-\frac{11}{5}\right)\).
Slope of \(CA'=\frac{5+\frac{11}{5}}{3-\frac{13}{5}}=\frac{36/5}{2/5}=18\).
Line BC: \(y-5=18(x-3)\Rightarrow 18x-y=49\).
Answer: (3) \(18x-y=49\)
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