If a point $\mathrm{R}(4, \mathrm{y}, \mathrm{z})$ lies on the line segment joining the points $\mathrm{P}(2,-3,4)$ and $\mathrm{Q}(8,0,10)$, then the distance of $R$ from the origin is:

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Three Dimensional Geometry

If a point

R (4, y, z)

lies on the line segment joining the points

P (2, - 3, 4)

and

Q (8, 0, 10)

, then the distance of

R

from the origin is:

(A) $53$
(B)6
(C) $214$ ✓ Correct
(D) $221$

Answer: (C)

Solution

Parametrise the segment from

P (2, - 3, 4)

Q (8, 0, 10)

\frac{x - 2}{6} = \frac{y + 3}{3} = \frac{z - 4}{6}

. The point

R

has

x = 4

, so the common parameter is

\frac{4 - 2}{6} = \frac{1}{3}

. Then

y + 3 = 3 \cdot \frac{1}{3} = 1

gives

y = - 2

, and

z - 4 = 6 \cdot \frac{1}{3} = 2

gives

z = 6

, so

R = (4, - 2, 6)

. Its distance from the origin is

4^{2} + (- 2)^{2} + 6^{2} = 16 + 4 + 36 = 56 = 214

JEE Main · 2019 · 08 Apr 2019 · Shift 2 · Single correct option

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