Question

În reperul cartezian xOy se considera punctele A(-1,a) B(0,-3) C(1,1). Determinati-l pe a știind ca AB+BC=AC

Answer 1

$A(-1,a),\quad B(0,-3),\quad C(1,1) \\ \\ AB+BC = AC \\ \\ \Rightarrow \text{A, B, C sunt coliniare.} \\ \Rightarrow \text{A, B, C apartin aceleiasi drepte.} \\ \Rightarrow A(-1,a) \text{ apartine dreptei BC} \\ \\ \begin{array}{rcl}BC&:& \quad \left|\begin{array}{ccc}x&y&1\\0&-3&1\\1&1&1\end{array}\right| = 0 \\ \\& & -3x+y+3-x = 0 \\ & & 4x - y - 3 = 0 \end{array} \\ \\ \\ A(-1,a) \in BC \\ \Rightarrow 4\cdot (-1) - a - 3 = 0 \Rightarrow -7 - a = 0 \Rightarrow \boxed{a = -7}$

Answer 2

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