First slide

Introduction to Determinants

Question

Suppose $a, b$ and $c$ are distinct real numbers. Let $Δ = |\begin{array}{ccc} a & a + c & a - b \\ b - c & b & a + b \\ c + b & c - a & c \end{array}| = 0$ .Then the straight line $a (x - 5) + b (y - 2) + c = 0$ passes through the fixed point

Moderate

A

(5, 2)
B

(6, 2)
C

(6, 3)
D

(5, 3)

Solution

Applying $C_{1} \to a C_{1} + b C_{2} + c C_{3}$ we get
Let $Δ = \frac{1}{a} |\begin{array}{ccc} a (a + b + c) & a + c & a - b \\ b (a + b + c) & b & a + b \\ c (a + b + c) & c - a & c \end{array}|$
$= \frac{1}{a} a (a + b + c) Δ_{1},$ where
$Δ_{1} = |\begin{array}{ccc} a & a + c & a - b \\ b & b & a + b \\ c & c - a & c \end{array}|$
Using $C_{2} \to C_{2} - C_{1}, C_{3} \to C_{3} - C_{1}$ we get
$\begin{array}{l} Δ_{1} = |\begin{array}{ccc} a & c & - b \\ b & 0 & a \\ c & - a & 0 \end{array}| = a (a^{2} + b^{2} + c^{2}) \\ ∴ Δ = (a + b + c) (a^{2} + b^{2} + c^{2}) = 0 \end{array}$
As $a, b, c$ are distinct real numbers, $a^{2} + b^{2} + c^{2} \neq 0$ therefore
$a + b + c = 0$
$\Rightarrow$ the line $a (x - 5) + b (y - 2) + c = 0$ passes through $(6, 3)$

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