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Chapter 14: Problem 1209

For the collinear points \(\mathrm{P}-\mathrm{A}-\mathrm{B}, \mathrm{AP}=4 \mathrm{AB}\), then \(\mathrm{P}\) divides AB from \(\mathrm{A}\) in the ratio \(\ldots \ldots .\) (a) \(4: 5\) (b) \(-4: 5\) (c) \(-5: 4\) (d) \(-1: 4\)

Short Answer

Expert verified
The correct ratio in which point P divides the line segment AB is \((-1:4)\), which corresponds to option (d).

Step by step solution

01

Understand the section formula for collinear points

For any two collinear points A(x1, y1) and B(x2, y2), if a point P(x, y) divides AB in the ratio m:n, then we can use the section formula to find the coordinates of point P: x = \(\frac{m * x2 + n * x1}{m + n}\) y = \(\frac{m * y2 + n * y1}{m + n}\)
02

Identify the given information

We are given that AP = 4AB. Let's denote the ratio in which P divides AB as m:n.
03

Apply the given information to the section formula

We know that AP = 4AB, so we can write this in terms of the ratio m:n as: m = 4*n Now we can use this equation to find the correct answer among the given options.
04

Check each option

(a) \(4: 5\): If the ratio is \(4:5\), then m = 4 and n = 5. Checking the equation, 4 = 4*5, which is false. (b) \(-4: 5\): If the ratio is \(-4:5\), then m = -4 and n = 5. Checking the equation, -4 = 4*5, which is also false. (c) \(-5: 4\): If the ratio is \(-5:4\), then m = -5 and n = 4. Checking the equation, -5 = 4*4, which is false. (d) \(-1: 4\): If the ratio is \(-1:4\), then m = -1 and n = 4. Checking the equation, -1 = 4*4, which is indeed true.
05

State the final answer

The correct ratio in which point P divides the line segment AB is \(\boxed{(-1:4)}\), which corresponds to option (d).

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