question_answer

Find the relation between x and y if the points \[A(x,y),\,B(-5,7)\] and \[C(-4,5)\]are collinear.

Answer:

Given that the points \[A(x,y),\,B(-5,7)\]and \[C(-4,5)\] are collinear.

So, the area formed by the vertices are 0.

Therefore,

\[\frac{1}{2}[{{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})]=0\]

\[\Rightarrow \frac{1}{2}[x(7-5)-5(5-y)-4(y-7)]=0\]

\[\Rightarrow \frac{1}{2}[x(2)-5(5-y)-4(y-7)]=0\]

\[\Rightarrow 2x-25+5y-4y+28=0\]

\[\Rightarrow 2x+y+3=0\]

\[-2x-3=y\]

which is the required, relation between x and y i.e., \[y=-2x-3\].