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The points A, B and $C$ are collinear and $A B = B C$ are collinear and . If the coordinates of A, B and $C$ . If the coordinates of A, B and are $(3, a), (1, 3)$ are and $(b, 4)$ and respectively, then what is the values of $a$ respectively, then what is the values of and $b$ and?

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2 and 4

-3 and 5

3 and 1

2 and -1

answer is D.

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Detailed Solution

Given data,
Three collinear points- A(3, a), B(1, 3) and C(b, 4).
AB = BC.
For the coordinates of the 2 points

x, y, m, n

respectively, the coordinates of the middle point of the two points will be,

x + m 2, y + n 2

If the points A, B, C are collinear, then,

A B = B C

. We know, the coordinates of A, B and C are

3, a, 1, 3, b, 4

. Calculating the value of a and b.
As

A B = B C

, thus point B is the middle point of A and C,
So, the coordinates of point B will be

3 + b 2, a + 4 2

. We know, the coordinates of point B is (1,3).
Hence,

3 + b 2 = 1 and a + 4 2 = 3

Now,

\frac{3 + b}{2} = 1

Equating b,

\frac{3 + b}{2} = 1

b = - 1

Similarly,

a + 4 2 = 3

Equating a,

\frac{a + 4}{2} = 3

a = 6 - 4

a = 2

Therefore, the value of a and b is 2 and -1 respectively.
Hence, option 4 is correct.

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