Point P divides the line segment joining the points A (-1, 3) and B (9, 8), such that $\frac{AP}{BP} = \frac{k}{1}$. If P lies on the line $x - y + 2 = 0$, then the value of k will be -

1. A

   4/3

2. B

   3/4

3. C

   2/3

4. D

   3/2

Appeared in: UPPSC 2023

Explanation

• The problem requires finding an unknown ratio 'k' by combining the section formula with the equation of a line.
• First, the coordinates of the dividing point P are expressed in terms of 'k' using the section formula: P = ((9k - 1)/(k + 1), (8k + 3)/(k + 1)).
• Since point P lies on the line x - y + 2 = 0, its coordinates must satisfy this equation.
• Substituting the coordinates of P into the line equation yields a linear equation in 'k': ((9k - 1)/(k + 1)) - ((8k + 3)/(k + 1)) + 2 = 0.
• Solving this equation by simplifying the terms (9k - 1) - (8k + 3) + 2(k + 1) = 0 leads to 3k - 2 = 0.
• The final result is k = 2/3.

Why Other Options Were Wrong

• Option A: This value would result from an algebraic error during the simplification of the linear equation after substitution.
• Option B: This value is the reciprocal of the correct answer (1/k). It likely arises from an inversion error at the final step, such as incorrectly solving 3k = 2 as k = 3/2 and then inverting it.
• Option D: This value is the reciprocal of the correct answer. It likely results from a sign error during simplification (e.g., getting 2k - 3 = 0 instead of 3k - 2 = 0) or an inversion error when solving for k.

Related Visual

Clinical Relevance

• Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Application of the section formula in coordinate geometry as background academic context rather than a clinical decision trigger.
• This question tests fundamental concepts in coordinate geometry, which is a branch of mathematics.
• While not directly used in clinical nursing practice, the logical reasoning and problem-solving skills are transferable to various analytical tasks.
• What if the point P divided the segment externally? The section formula would change to x = (m₁x₂ - m₂x₁) / (m₁ - m₂) and y = (m₁y₂ - m₂y₁) / (m₁ - m₂), leading to a different value for k.

How to Approach the Question

• Identify the given information: coordinates of points A and B, the division ratio k:1, and the equation of the line that point P lies on.
• Recall the section formula to write the coordinates (x, y) of the dividing point P in terms of the unknown variable 'k'.
• Recognize that if a point lies on a line, its coordinates must satisfy the line's equation.
• Substitute the x and y coordinates of P into the given line equation, x - y + 2 = 0.
• Solve the resulting algebraic equation for the variable 'k'.
• Compare the calculated value of 'k' with the given options to find the correct answer.

Concept Tested & Keywords

• Concept Tested: Application of the section formula in coordinate geometry.
• Stem keywords: divides, line segment, A (-1, 3), B (9, 8), ratio, k/1, lies on the line, x - y + 2 = 0
• Lead-in keywords: value of k

Question ID

Q4_fb1QKHpN0iUVnhfNK9d

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