First slide

Equation of line in Straight lines

Question

$If A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3}) are the vertices of a triangle, then the equation |\begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix}| + |\begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{3} & y_{3} & 1 \end{matrix}| =0$ represents

Difficult

A

the altitude throught A
B

the perpendicular bisector of side BC
C

the median through A
D

None of these

Solution

The given equation becomes
$|\begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} + x_{3} & y_{2} + y_{3} & 2 \end{matrix}| = 0 \Rightarrow |\begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ \frac{x_{2} + x_{3}}{2} & \frac{y_{2} + y_{3}}{2} & 1 \end{matrix}| = 0$
It is the equation of the line passing through A and mid point of BC.

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