Question

determine if (3,7),(4,5) and (2,11) are collinear or not ​

Answer 1

Answer:

Question :-

determine the points (1,5),(2,3) and (-2,11) are collinear.

Formula used :-

For collinearity of three points ,

\: \boxed{ 0 = \small \frac{1}{2} \big( \: x_1(y_2 - y_3) + x_2(y_3 - y_1) +x_3(y_1 - y_2) \big)}

0=

2

1

(x

1

(y

2

−y

3

)+x

2

(y

3

−y

1

)+x

3

(y

1

−y

2

))

_______________________________

Solution :-

Given that :-

Given points are ( 1,5) , (2,3) and (-2,11)

\begin{gathered}let \\ x_1 = 1 ,\: y_1 = 5 \\ \\ x_2 = 2 \: , \: y_2 = 3 \\ \\ x_3 = - 2 \: , \: y_3 = 11\end{gathered}

let

x

1

=1,y

1

=5

x

2

=2,y

2

=3

x

3

=−2,y

3

=11

Now using the given formula ,

\begin{gathered}\small \: 0 = \frac{1}{2} \big(1(3 - 11) + 2(11 - 5) - 2(5 - 3) \big) \\ \\ \implies \: 0 = \frac{1}{2} \big( - 8 + 12 - 4) \\ \\ \implies \: 0 = \frac{1}{2} ( - 12 + 12) \\ \\ \implies \: \boxed{0 = 0}\end{gathered}

0=

2

1

(1(3−11)+2(11−5)−2(5−3))

⟹0=

2

1

(−8+12−4)

⟹0=

2

1

(−12+12)

⟹

0=0

Hence proved,

The given points are collinear.