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Mathematics
Problem
Which one of the following conditions guarantees that the three points
(
x
1
,
y
1
)
(x_1,y_1)
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
(x_2,y_2)
(
x
2
,
y
2
)
and
(
x
3
,
y
3
)
(x_3,y_3)
(
x
3
,
y
3
)
are collinear? a)
x
1
y
2
+
x
2
y
3
+
x
3
y
1
=
0
x_1y_2+x_2y_3+x_3y_1=0
x
1
y
2
+
x
2
y
3
+
x
3
y
1
=
0
b)
y
2
−
y
1
x
2
−
x
1
+
y
3
−
y
2
x
3
−
x
2
=
0
\displaystyle \frac{y_2-y_1}{x_2-x_1}+\frac{y_3-y_2}{x_3-x_2}=0
x
2
−
x
1
y
2
−
y
1
+
x
3
−
x
2
y
3
−
y
2
=
0
c)
∣
x
1
y
1
1
x
2
y
2
1
x
3
y
3
1
∣
=
0
\begin{vmatrix}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{vmatrix}=0
x
1
x
2
x
3
y
1
y
2
y
3
1
1
1
=
0
d)
∣
x
1
y
1
x
2
y
2
∣
=
0
\begin{vmatrix}x_1 & y_1 \\ x_2 & y_2\end{vmatrix}=0
x
1
x
2
y
1
y
2
=
0
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