First slide

Equation of line in Straight lines

Question

The area of the triangle with vertices at $(a, b + c), (b, c + a)$ and $(c, a + b)$ , is

Moderate

A

0
B

a+b+c
C

ab+bc+ca
D

none of these

Solution

Let $∆$ be the area of the triangle with vertices at $(a, b + c), (b, c + a)$ and $(c, a + b)$ , Then,
$∆$ = absolute value of $\frac{1}{2} |\begin{array}{lll} a b + c 1 \\ b c + a 1 \\ c a + b 1 \end{array}|$
$\Rightarrow$ $∆$ = Absolute value of $f \frac{1}{2} |\begin{array}{lll} a a + b + c 1 \\ b b + c + a 1 \\ c c + a + b 1 \end{array}| [\begin{matrix} Applying \\ C_{2} \to C_{2} + C_{1} \end{matrix}]$
$\Rightarrow$ $∆$ = Absolute value of $\frac{1}{2} (a + b + c) |\begin{array}{lll} a 1 1 \\ b 1 1 \\ c 1 1 \end{array}|$
$[Taking a + b + c common from C_{2}]$
$\Rightarrow Δ = 0$

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