Question

Let D be harmonic conjugate of point C with respect to points \[ A(1,-3,5),\qquad B(5,-3,1) \] If C divides AB in ratio \(3:5\), then point dividing CD in ratio \(1:2\) is

• A. \((-5,-3,11)\)
• B. \(\left(\frac52,-3,\frac72\right)\)
• C. \((3,-3,3)\)
• D. \((0,-3,6)\)

Hint:

In harmonic division, if one point divides internally in ratio \(m:n\), harmonic conjugate divides externally in same ratio.

Answer 1

The Correct Option is C

Concept: Harmonic division means \[ (A,B;C,D)=-1 \] If C divides internally in ratio \(m:n\), then D divides externally in same ratio.

Step 1: Find coordinates of C.
Using section formula: \[ C= \left( \frac{3(5)+5(1)}8, -3, \frac{3(1)+5(5)}8 \right) \] \[ = \left( \frac{20}{8}, -3, \frac{28}{8} \right) \] \[ = \left( \frac52,-3,\frac72 \right) \]

Step 2: Find harmonic conjugate D.
External division same ratio. \[ D=(-5,-3,11) \]

Step 3: Point dividing CD in ratio \(1:2\).
Using section formula: \[ P= \left( \frac{1(-5)+2(\frac52)}3, \frac{1(-3)+2(-3)}3, \frac{1(11)+2(\frac72)}3 \right) \] \[ = (3,-3,3) \] Thus \[ \boxed{(3,-3,3)} \]