Question

If (1,2),(-1,b),(-3,-4) are collinear,then find the value of b.

Answer 1

Answer:

4

thanks for the update and for the4 5

Answer 2

Answer:

The value of b is -1.

Step-by-step explanation:

ATQ, the points (1, 2), (-1, b) & (-3, -4) are collinear.

We know that when three collinear points are joined through straight lines, the area enclosed by them will be equal to 0.

Therefore, we'll use the area of a triangle (formed by coordinates) formula to find out the value of 'b'.

$\boxed{\boxed{\sf \ Area \ of \ \triangle = \frac{1}{2} \Bigg( \ x_1\left(y_2 - y_3\right) + x_2\left(y_3 - y_1\right) + x_3\left(y_1 - y_2\right)\Bigg) \ }}$

ATQ,

x₁ $\longmapsto$ 1

x₂ $\longmapsto$ -1

x₃ $\longmapsto$ -3

y₁ $\longmapsto$ 2

y₂ $\longmapsto$ b

y₃ $\longmapsto$ -4

Now, Since we know that they area formed by joining the points is 0, we'll equate the area of a triangle formula with 0.

$\Longrightarrow \sf Area \ of \ \triangle = 0\\ \\ \\\Longrightarrow \sf \dfrac{1}{2} \Bigg(x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\Bigg)}} = 0\\ \\ \\ \\\Longrightarrow \sf \dfrac{1}{2} \Bigg(1(b - (-4)) + -1(-4 - 2) + (- 3)(2 - b)\Bigg)}} = 0\\ \\ \\ \\\Longrightarrow \sf \dfrac{1}{2} \Bigg(1(b + 4) + -1(-6) - 3(2 - b)\Bigg)}} = 0$

$\Longrightarrow \sf \dfrac{1}{2} \Bigg(b + 4 + 6 - 6 + 3b\Bigg)}} = 0\\ \\ \\ \\\Longrightarrow \sf 4b + 4 = 0 \times 2\\ \\ \\ \\\Longrightarrow \sf 4b = -4\\ \\ \\ \\\Longrightarrow \sf b = -1$

∴ The value of b is -1.