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Conic Sections Ncert Solutions Maths class 11th Maths Class 11th

Let O be the origin. Let OP = xi + yj - k and OQ = -i + 2j + 3xk, x, y ∈ R, x > 0, be such that |PQ| = √20 and the vector OP is perpendicular to OQ. If OR = 3i + zj - 7k, z ∈ R, is coplanar with OP and OQ, then the value of x² + y² + z² is equal to:

Option 1 -

Option 2 -

Option 3 -

Option 4 -

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Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

Given vectors OP = xi + yj - k and OQ = -i + 2j + 3xk.
PQ = OQ - OP = (-1 - x)i + (2 - y)j + (3x + 1)k
Given |PQ| = √20, so |PQ|² = 20.
(-1 - x)² + (2 - y)² + (3x + 1)² = 20
(1 + x)² + (2 - y)² + (3x + 1)² = 20 .(i)
- Given OP ⊥ OQ, so OP · OQ = 0.
  (x)(-1) + (y)(2) + (-1)(3x) = 0
  -x + 2y - 3x = 0 ⇒ -4x + 2y = 0 ⇒ y = 2x .(ii)
Substitute (ii) into (i):
(1 + x)² + (2 - 2x)² + (3x + 1)² = 20
1 + 2x + x² + 4 - 8x + 4x² + 9x² + 6x + 1 = 20
14x² = 14 ⇒ x² = 1 ⇒ x = ±1.
When x = 1, y = 2. When x = -1, y = -2.
So, (x, y) can be (1, 2) or (-

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Let O be the origin. Let OP = xi + yj - k and OQ = -i + 2j + 3xk, x, y ∈ R, x > 0, be such that |PQ| = √20 and the vector OP is perpendicular to OQ. If OR = 3i + zj - 7k, z ∈ R, is coplanar with OP and OQ, then the value of x² + y² + z² is equal to:

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