If the points P(1, 2), B(0, 0) and C(a, b) are collinear, prove
Answers
Answer:
The relation between a and b is 2a=b. It means b is twice of a or a is half of b.
Step-by-step explanation:
The given points are A(1,2),B(0,0),C(a,b).
These points are collinear if slope of AB is equal to BC.
Slope formula:
m=\frac{y_2-y_1}{x_2-x_1}m=
x
2
−x
1
y
2
−y
1
The slope of AB is
m_1=\frac{0-2}{0-1}=2m
1
=
0−1
0−2
=2
The slope of BC is
m_2=\frac{b-0}{a-0}=\frac{b}{a}m
2
=
a−0
b−0
=
a
b
If A,B, and C are collinear, then
m_1=m_2m
1
=m
2
2=\frac{b}{a}2=
a
b
2a=b2a=b
Therefore the relation between a and b is 2a=b. It means b is twice of a or a is half of b.
Answer:
2a=b
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