Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point&nbsp;<math><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math>&nbsp;in the rotated plane in B(a, b, c), then

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Maths NCERT Exemplar Solutions Class 12th Chapter Six Probability Ncert Solutions Maths class 12th Maths Class 12th

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point $(2, - \frac{1}{2}, 2)$ in the rotated plane in B(a, b, c), then

Option 1 -

$\frac{a}{8} = \frac{b}{5} = \frac{c}{- 4}$

Option 2 -

$\frac{a}{4} = \frac{b}{5} = \frac{c}{- 2}$

Option 3 -

$\frac{a}{8} = \frac{b}{- 5} = \frac{c}{4}$

Option 4 -

$\frac{a}{4} = \frac{b}{5} = \frac{c}{2}$

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1 Answer

Answered by

Payal Gupta | Contributor-Level 10

2 months ago

Correct Option - 1

Detailed Solution:

Consider the equation of plane,

$P : (2 x + 3 y + z + 20) + λ (x - 3 y + 5 z - 8) = 0$

$?$ Plane P is perpendicular to 2x + 3y + z + 20 = 0

So, $4 + 2 λ + 9 - 9 λ + 1 + 5 λ = 0$

0 $λ = 7$

P : 9x – 18y + 36z – 36 = 0

Or P : x – 2y + 4z = 4

If image of

$(2, - \frac{1}{2}, 2)$

In plane P is (a, b, c) then

$\frac{a - 2}{1} = \frac{b + \frac{1}{2}}{- 2} = \frac{c - 2}{4}$

and $(\frac{a + 2}{2}) - 2 (\frac{b - \frac{1}{2}}{2}) + 4 (\frac{c + 2}{2}) = 4$

clearly

$a = \frac{4}{3}, b = \frac{5}{6} a n d c = - \frac{2}{3}$

So, a : b : c = 8 : 5 : 4

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Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point $(2, - \frac{1}{2}, 2)$ in the rotated plane in B(a, b, c), then

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Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point (2,−12,2) in the rotated plane in B(a, b, c), then

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Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x – 3y + 5z = 8. If the mirror image of the point $(2, - \frac{1}{2}, 2)$ in the rotated plane in B(a, b, c), then