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Answer 1

Question:

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

Answer:

$\huge\boxed{\boxed{\red{\bf{2x+y+3=0}}}}$

Step-by-step explanation:

We can approach this in 2 ways .

First method :-

$\mathbb{\underline{SLOPE}}$

$\mathtt{Slope=\frac{y_2-y_1}{x_2-x_1}}$

Since the 2 lines are collinear :

Slope of AB = Slope of BC .

$\dfrac{7-y}{-5-x}=\frac{5-7}{-4+5}\\\\\implies -\dfrac{7-y}{5+x}=\frac{-2}{1}\\\\\implies -1(7-y)=-2(5+x)\\\\\implies -7+y=-10-2x\\\\\implies 2x+y+3=0$

Second method :-

$\mathbb{\underline{AREA\:OF\:TRIANGLE}}$

Since the lines are collinear , area of triangle ABC = 0 .

$\dfrac{1}{2} |(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)| =0\\\\\implies |x(7-5)-5(5-y)-4(y-7)|=0\\\\\implies |2x - 25 + 5y - 4y + 28| = 0\\\\\implies 2x+y+3=0$

Hence the relation will be 2 x + y + 3 = 0 .

Answer 2

Answer:

such a thing happen gun then the flow of line ab is equals to slope of line BC