Question : The points $A(3,–2)$, $B(1,4)$, and $C(–2,x)$ are collinear. What is the value of $x$?
Option 1: 13
Option 2: –2
Option 3: 5
Option 4: 3
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Correct Answer: 13
Solution : Given: $A(3,–2)$, $B(1,4)$ and $C(–2,x)$ are collinear. The point $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ are collinear if $\frac{y_2–y_1}{x_2–x_1}=\frac{y_3–y_1}{x_3–x_1}$ Putting the values, we have, ⇒ $\frac{4–(–2)}{1–3}=\frac{x–(–2)}{(–2)–3}$ ⇒ $\frac{6}{–2}=\frac{x+2}{–5}$ ⇒ $–30=–2x–4$ ⇒ $2x=26$ ⇒ $x=13$ Hence, the correct answer is 13.
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