Question

Answer the given question ...​

Answer 1

$\large\underline{\sf{Solution-}}$

Given Coordinates are (1, 3), (7, - 3), (5, - 1) and (6, - 2)

Let assume that

Coordinates of A be (1, 3)

Coordinates of B be (7, - 3)

Coordinates of C be (5, - 1)

and

Coordinates of D be (6, - 2)

Let we first check the collinearity of points.

To check, we use method of slope.

We know,

Condition for three points to be collinear

Slope between two points (a, b) and (c, d) is given by

$\boxed{ \rm{ Slope = \frac{d - b}{c - a}}}$

and we know, three points A, B and C are collinear iff slope of AB = slope of BC.

Now,

$\red{\rm :\longmapsto\:Slope \: of \: AB = \dfrac{ - 3 - 3}{7 - 1} = \dfrac{ - 6}{6} = - 1}$

$\red{\rm :\longmapsto\:Slope \: of \: BC = \dfrac{1 - 3}{7 - 5} = \dfrac{ - 2}{2} = - 1}$

$\red{\rm :\longmapsto\:Slope \: of \: CD \: = \dfrac{ - 2 + 1}{6 - 5} = \dfrac{ - 1}{1} = - 1}$

$\red{\rm :\longmapsto\:Slope \: of \: AD = \dfrac{ - 2 - 3}{6 - 1} = \dfrac{ - 5}{5} - 1}$

Thus, we get

Slope of AB = Slope of BC, implies that AB || BC

Slope of BC = Slope of CD, implies that BC || CD

It implies, points A, B, C, D are collinear.

Hence,

One and only one line can passes through these 4 collinear points.

We know,

$\boxed{ \rm{ \: ^nC_n \: = \: 1}}$

So, it means,

$\boxed{ \rm{ Number \: of \: lines \: passes \: through\: 4 \: points \: = 1 = \: ^4C_4}}$

Hence,

• Option 4) is correct.