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Q.

The line segment joining the points $A (2, 1)$ and $B (5, − 8)$ is trisected at the points $P$ and $Q$ such that $P$ is nearer to $A$ . Find the coordinates of P and Q.

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a

P (3, 2) and Q = (4, − 5)

b

P (3, − 2) and Q = (4, 5)

c

P (3, 2) and Q = (4, 5)

d

P (3, − 2) and Q = (4, − 5)

answer is D.

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Detailed Solution

It is given that
P and Q are the points of trisection of the line segment joining the points

A (2, 1)

and

B (5, - 8)

.
We know that,
If points M and N divide the line segment XY into three equal parts, then M divides XY in the ratio of 1:2 and N divides XY in the ratio of 2:1.
Then point P divides the line AB in the ratio of 1:2 and Q divides the line AB in the ratio of 2:1.
Then,
We know that, if point

O

divides the line segment joining

M (x_{1}, y_{1}) and N (x_{2}, y_{2})

in the ratio

m_{1} : m_{2}

, then the coordinates of O will be

m 1 x 2 + m 2 x 1 m 1 + m 2, m 1 y 2 + m 2 y 1 m 1 + m 2

.
By using above section formula,
Coordinates of the point P is

P = 1 × 5 + 2 × 2 1 + 2, 1 × (− 8) + 2 × 1 1 + 2 ⇒ P = 5 + 4 3, − 8 + 2 3 ⇒ P = 93, − 6 3 ⇒ P = (3, − 2)

Similarly,
Coordinates of the point Q is

Q = 2 × 5 + 1 × 2 2 + 1, 2 × (− 8) + 1 × 1 2 + 1 ⇒ Q = 10 + 2 3, − 16 + 1 3 ⇒ Q = 123, − 15 3 ⇒ Q = (4, − 5)

Therefore,

P (3, − 2) and Q = (4, − 5)

.
Hence, option (4) is correct.

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