Question

Determine if the points 1, 5, 2, 3 and minus 2, 11 are collinear

Answer 1

$1 \div 2 |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|$
1/2|1(3-11)+2(11-5)+(-2)(5-3)|
=1/2|(-8)+12+(-4)|
=1/2(0)
=0
so,these points are collinear

Answer 2

$\bf\huge\boxed{\boxed{\bf\huge\:Hello\:Mate}}}$

$\bf\huge Let A(1 , 5) , \: B(2 , 3) , \: C(-2 , -11)$

$\bf\huge AB= \sqrt{(2 - 1)^2 + (3 - 5)^2}$

$\bf\huge = \sqrt{1^2 + (-2)^2}$

$\bf\huge = \sqrt{1 + 4} = \sqrt{5}$

$\bf\huge BC = \sqrt{(-2 - 2)^2 + (-11 - 3)^2}$

$\bf\huge BC = \sqrt{(-4)^2 + (-14)^2}$

$\bf\huge BC = \sqrt{16 + 196}$

$\bf\huge BC = \sqrt{212}$

$\bf\huge AB = \sqrt{4\times 53} = 2\sqrt{53}$

$\bf\huge AC = \sqrt{(-2 - 1)^2 + ( - 11 - 5)^2}$

$\bf\huge AC = \sqrt{(-3)^2 + (- 16)^2}$

$\bf\huge AC = \sqrt{9 + 256} = \sqrt{265}$

$\bf\huge Hence \:A , B\: and \:C\: are\: not\: collinear$

$\bf\huge\boxed{\boxed{\:Regards=\:Yash\:Raj}}}$