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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