3D Geometry
Section formula and collinearity
Grade None
Question:
<p>The general coordinates of a point <em>R</em> which divides the line joining <em>A</em>(3, –2, 4) and <em>B</em>(1, 1, 1) in the ratio <em>μ</em> : 1 are <br>\(\left(\dfrac{\mu+3}{\mu+1},\ \dfrac{\mu-2}{\mu+1},\ \dfrac{\mu+4}{\mu+1}\right)\). If <em>C</em>(–1, 4, –2) lies on the line <em>AB</em>, then the value of |<em>μ</em>| is:</p>
<p>1</p>
<p>2</p>
<p>3</p>
<p>4</p>
Step-by-Step Solution
Key Concept: If point C lies on line AB, then C must satisfy the section formula coordinates for some value of μ. Equating C's coordinates with the general point R will yield the value of μ.
Step 1: Since C(–1, 4, –2) lies on line AB, it must be expressible using the section formula. Therefore: –1 = (μ + 3)/(μ + 1) 4 = (μ – 2)/(μ + 1) –2 = (μ + 4)/(μ + 1) Step 2: Using the first equation: –1(μ + 1) = μ + 3 –μ – 1 = μ + 3 –2μ = 4 μ = –2 Step 3: Verify with the second equation: 4 = (–2 – 2)/(–2 + 1) = –4/(–1) = 4 ✓ Step 4: Verify with the third equation: –2 = (–2 + 4)/(–2 + 1) = 2/(–1) = –2 ✓ Step 5: Therefore, |μ| = |–2| = 2 ∴ Answer: B
Correct Answer: B