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JEE Main 202621 JanuaryEvening Shift
Exam
JEE Main
Year
2026
Date
21 January
Shift
Evening Shift
Subject
Mathematics
Chapter
Three Dimensional Geometry

JEE Main 2026 · 21 January · Evening Shift · Mathematics · Three Dimensional Geometry

Let the line L pass through the point (-3,5,2) and make equal...

Let the line LL pass through the point (−3,5,2)(-3,5,2) and make equal angles with the positive coordinate axes. If the distance of L from the point (−2,r,1)(-2, \mathrm{r}, 1) is 143\sqrt{\frac{14}{3}}, then the sum of all possible values of r is :

Options

  1. A10
  2. B6
  3. C12
  4. D16

Correct answer

A. 10

Step-by-step solution

Line LL through (−3,5,2)(-3, 5, 2) making equal angles with axes has direction (1,1,1)(1, 1, 1).
Point Q=(−2,r,1)Q = (-2, r, 1), PQ⃗=(1,r−5,−1)\vec{PQ} = (1, r-5, -1).
PQ⃗×d⃗=(r−4,−2,6−r)\vec{PQ} \times \vec{d} = (r-4, -2, 6-r).
∣PQ⃗×d⃗∣2=(r−4)2+4+(6−r)2=2r2−20r+56|\vec{PQ} \times \vec{d}|^2 = (r-4)^2 + 4 + (6-r)^2 = 2r^2 - 20r + 56.
Distance =2r2−20r+563=143= \sqrt{\frac{2r^2 - 20r + 56}{3}} = \sqrt{\frac{14}{3}}.
2r2−20r+56=14⇒r2−10r+21=0⇒r=32r^2 - 20r + 56 = 14 \Rightarrow r^2 - 10r + 21 = 0 \Rightarrow r = 3 or r=7r = 7.
Sum =3+7=10= 3 + 7 = 10.
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