Question 15
If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then
(A) AP = 1/3 AB (B) AP = PB (C) PB = 1/3 AB (D) AP = 1/2 AB
We need to find ratio between AP & PB
Let the ratio be k : 1
Also,
x1 = 4, y1 = 2
x2 = 8, y2 = 4
& x = 2, y = 1
Using section formula
𝒙 = (𝒎𝟏 𝒙𝟐 + 𝒎𝟐 𝒙𝟏)/(𝒎𝟏 + 𝒎𝟐)
2 = (𝑘 × 8 + 1 × 4)/(𝑘 + 1)
2 = (8𝑘 + 4)/(𝑘 + 1)
2(k + 1) = 8k + 4
2k + 2 = 8k + 4
2 − 4 = 8k − 2k
−2 = 6k
6k = − 2
k = (−2)/6
k = (−𝟏)/𝟑
Since the ratio is negative,
Point P divides AB externally
And ratio is 1 : 3
So, our figure looks like
Thus,
𝐴𝑃/𝑃𝐵 = 1/3
𝑨𝑷=𝟏/𝟑 𝑷𝑩
Given options are:
(A) AP = 1/3 AB (B) AP = PB (C) PB = 1/3 AB (D) AP = 1/2 AB
So, it does not match any of the options
𝐴𝑃/𝑃𝐵 = 1/3
Putting PB = PA + AB
𝑨𝑷/(𝑨𝑷 + 𝑨𝑩) = 𝟏/𝟑
3AP = AP + AB
3AP − AP = AB
2AP = AB
AP = 𝟏/𝟐 AB
So, the correct answer is (D)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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