# If the points $A (-1, -4), B (b, c)$ and $C (5, -1)$ are collinear and $2b + c = 4$, find the values of $b$ and $c$.

Given:

Points $A (-1, -4), B (b, c)$ and $C (5, -1)$ are collinear and $2b + c = 4$.

To do:

We have to find the values of $b$ and $c$.

Solution:

Let $A (-1, -4), B (b, c)$ and $C (5, -1)$ be the vertices of $\triangle ABC$.

$2b+c=4$

$\Rightarrow c=4-2b$......(i)

We know that,

If the points $A, B$ and $C$ are collinear then the area of $\triangle ABC$ is zero.

Area of a triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ is given by,

Area of $\Delta=\frac{1}{2}[x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2})]$

Therefore,

Area of triangle $ABC=\frac{1}{2}[-1(c+1)+b(-1+4)+5(-4-c)]$

$0=\frac{1}{2}[-c-1+3b-20-5c]$

$0(2)=(3b-6c-21)$

$0=3(b-2c-7)$

$b-2c=7$

$b-2(4-2b)=7$       (From (i))

$b-8+4b=7$

$5b=7+8$

$b=\frac{15}{5}$

$b=3$

$c=4-2(3)=4-6=-2$

The values of $b$ and $c$ are $3$ and $-2$ respectively.

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Updated on: 10-Oct-2022

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