21. If the points A(2, 3), B(5, k) and C(6, 7) are collinear then
(a) a = b
(b) a = 2b
(c) 2a = b
(d) a + b = 0
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Answer:
Since the given points are collinear, they do not form a triangle, which means
area of the triangle is Zero.
Area of a triangle with vertices (x
1
,y
1
) ; (x
2
,y
2
) and (x
3
,y
3
) is
∣
∣
∣
∣
∣
2
x
1
(y
2
−y
3
)+x
2
(y
3
−y
1
)+x
3
(y
1
−y
2
)
∣
∣
∣
∣
∣
Hence, substituting the points (x
1
,y
1
)=(1,2) ; (x
2
,y
2
)=(0,0) and (x
3
,y
3
)=(a,b)
in the area formula, we get
∣
∣
∣
∣
∣
2
1(0−b)+0(a−1)+a(2−0
∣
∣
∣
∣
∣
=0
=>−b+2a=0
=>2a=b
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