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Point P divides the line segment joining the points $A (2, 1)$ and $B (5, − 8)$ such that $A P A B = 13$ . If P lies on the line $2 x − y + k = 0$ , find the value of $k$ .

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k = − 214

k = − 234

k = − 254

k = − 274

answer is D.

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Detailed Solution

Given points are

A (2, 1)

and

B (5, − 8)

.
Since,

A P A B = 13

.
So, the point P divides the line segment AB in the ratio of

1 : 3

.
We know that,
The coordinates of the point which divides a line joining two points

x 1, y 1

and

x 2, y 2

in the ratio m:n is

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

.
Here,

m = 1 and n = 3

.
Then,
Coordinate of P is,

P 1 × 5 + 3 × 2 1 + 3, 1 × (− 8) + 3 × 1 1 + 3 P 5 + 6 4, − 8 + 3 4 P 114, − 5 4 .

Since, point P lies on the given line

2 x − y + k = 0

.
Therefore,

2 x − y + k = 0 ⇒ 2 × 114 − − 54 + k = 0

⇒ 224 + 54 + k = 0 ⇒ k + 274 = 0 ⇒ k = − 274

The value of

k = − 274

.
Hence, option (4) is correct.

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