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The assertion in this question is followed by a reason.

Assertion: The coordinates of the points which divide the line segment joining $A (4, − 1)$ and $B (− 2, − 3)$ andinto three equal parts $2, − 5 3$ into three equal parts and $0, − 7 3$ and.

Reason: The points which divide AB in the ratio 1: 3 and 3: 1 are called points of trisection of AB.

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If both Assertion and Reason are correct and Reason is the correct explanation of Assertion.

If both Assertion and Reason are correct, but Reason is not the correct explanation of Assertion.

If Assertion is correct but Reason is incorrect.

If Assertion is incorrect but Reason is correct.

answer is C.

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Detailed Solution

We have been given,

A = (4, − 1) B ≡ (− 2, − 3)

First, considering m:n = 1: 2.
Using the section formula,

(x, y) = (\frac{{mx}_{2} + {nx}_{1}}{m + n}, \frac{{my}_{2} + {ny}_{1}}{m + n})

Substituting

m = 1, x_{2} = - 2, n = 2, x_{1} = 4, y_{2} = - 3, and y_{1} = - 1

in the section formula,

1 (− 2) + 2 (4) 1 + 2, 1 (− 3) + 2 (− 1) 1 + 2 = 2, − 53

Now consider

m : n = 2 : 1

.
Substituting

m = 2, x_{2} = - 2, n = 1, x_{1} = 4, y_{2} = - 3, and y_{1} = - 1

in the section formula,
So, the point of intersection is,

2 (− 2) + 1 (4) 1 + 2, 2 (− 3) + 1 (− 1) 1 + 2 ≡ 0, − 73

Thus, the assertion is correct.
The point that splits the segment in the ratio 1:2 is now referred to as a point of trisection.
Therefore, the reason is false.
Hence, option 3 is correct.

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