Question

Find the value of for which the points (–5, 1), (1,) and (4, –2) are collinear.

Answer 1

Step-by-step explanation:

Given -

• (-5,1), (1,k), (4,-2) are collinear

To Find -

• Value of k

Method 1 :-

As we know that :-

• x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)

→ (-5,1) = (x1,y1)

(1,k) = (x2,y2)

(4,-2) = (x3,y3)

Now,

→ -5(k + 2) + 1(-2 - 1) + 4(1 - k) = 0

→ -5k - 10 - 3 + 4 - 4k = 0

→ 9k = -9

→ k = -1

Method 2 :-

As we know that :-

• Slope = y2 - y1/x2 - x1

Now,

Let a = (-5,1), b = (1,k), c = (4,-2)

Then,

Slope of line AB =

(-5,1) = (x1,y1)

(1,k) = (x2, y2)

→ k - 1/1 + 5

→ k - 1/6

And

Slope of line BC =

(1,k) = (x1,y1)

(4,-2) = (x2,y2)

→ -2 - k/4 - 1

→ -2 - k/3

Now,

→ k - 1/6 = -2 - k/3

→ 3(k - 1) = 6(-2 - k)

→ 3k - 3 = -12 - 6k

→ 9k = -12 + 3

→ 9k = -9

→ k = -1

Hence,

The value of k is -1

Answer 2

$\huge \mathfrak{Answer:-}$

$\large\underline\mathrm{The \: value \: of \: k \: is \: -1.}$

$\large\underline\mathrm{Given:-}$

• (–5, 1), (1, k) and (4, –2) are collinear.

$\large\underline\mathrm{To \: find}$

• value of k ?

$\large\underline\mathrm{A/Q.}$

• x1(y2 - y3) + x2 (y3 - y1) + x3(y1 - y2)

$\implies$ x1 = -5. y1 = 1

$\implies$ x2 = 1. y2 = k

$\implies$ x3 = 4. y3 = -3

$\large\underline\mathrm{Solution}$

$\implies$ -5(k - 2) + 1(-2 - 1) + 4(1 - k) = o

$\implies$ -5k - 10 - 3 + 4 - 4k = 0

$\implies$ 9k = -9

$\implies$ k = -1

slope =y2 - y1 / x2 - x1

$\large\underline\mathrm{Now}$

Let a = (-5, 1), b = (1, k), c = (4, -2)

$\large\underline\mathrm{Then,}$

$\large\underline\mathrm{slope \: of \: line \: AB}$

$\implies$ (-5, 1) = (x1, y1)

$\implies$ (1, - k) = (x2 - x1)

$\implies$ k = 1/1 + 5

$\implies$ k = 1/6

$\large\underline\mathrm{and}$

$\large\underline\mathrm{slope \: of \: line \: BC}$

$\implies$ (1, k) = (x2, y2)

$\implies$ (4, - 2) = (x3 - x3)

$\implies$ -2 - k/4 - 1

$\implies$ -2 - k/3

$\large\underline\mathrm{Now}$

$\implies$ k - 1/6 = -2 - k/3

$\implies$ 3(k - 1) = 6(-2 - k)

$\implies$ 3k - 3 = -12 - 6k

$\implies$ 9k = -12 + 3

$\implies$ 9k = -9

$\implies$ k = -1

$\large\underline\mathrm{hence,}$

$\large\underline\mathrm{The \: value \: of \: k \: is \: -1.}$

$\large\underline\mathrm{Hope \: it \: helps \: you \: plz \: mark \: me \: brainlist}$