Question

Let \( P(a, 4, 7) \) and \( Q(3, \beta, 8) \) be two points. If the YZ-plane divides the join of the points P and Q in the ratio 2:3 and the ZX-plane divides the join of P and Q in the ratio 4:5, then the length of line segment PQ is:

• A. \( \sqrt{107} \)
• B. \( \sqrt{27} \)
• C. \( \sqrt{83} \)
• D. \( \sqrt{97} \)

Hint:

For 3D coordinate division problems, apply the section formula separately for each coordinate. The YZ-plane forces \( x=0 \) and the ZX-plane forces \( y=0 \), helping to determine unknowns.

Answer 1

The Correct Option is A

Step 1: Using section formula for the YZ-plane Since the YZ-plane divides the line segment in the ratio \(2:3\), its x-coordinate must be 0. Using the section formula for x-coordinate: \[ x = \frac{3a + 2(3)}{3+2} = 0 \] \[ \frac{3a + 6}{5} = 0 \] \[ 3a + 6 = 0 \] \[ a = -2 \] Step 2: Using section formula for the ZX-plane Since the ZX-plane divides the line segment in the ratio \(4:5\), its y-coordinate must be 0. Using the section formula for y-coordinate: \[ y = \frac{5(4) + 4\beta}{5+4} = 0 \] \[ \frac{20 + 4\beta}{9} = 0 \] \[ 20 + 4\beta = 0 \] \[ \beta = -5 \] Step 3: Finding the length of PQ Now that we have \( P(-2, 4, 7) \) and \( Q(3, -5, 8) \), we use the distance formula: \[ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] \[ = \sqrt{(3 - (-2))^2 + (-5 - 4)^2 + (8 - 7)^2} \] \[ = \sqrt{(3 + 2)^2 + (-9)^2 + (1)^2} \] \[ = \sqrt{5^2 + 9^2 + 1^2} \] \[ = \sqrt{25 + 81 + 1} \] \[ = \sqrt{107} \]