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,

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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