Question

22 A line having direction ratios 1,2,2 cut two
planes
3x - 2y +62 +12 = 0,
3x - 2y + 6z - 2 = () at
P and Q respectively.If PQ is 2 then
[22] =((.) denotes greatst integer function)​

Answer 1

Answer:

pq is 2 then (2 ×lambda ) =4

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Answer 2

[2PQ] = 9

Step-by-step explanation:

Given:

Two planes 3x - 2y + 6z + 12 = 0 and 3x - 2y + 6z - 2 = 0

A line having direction ratios 1, 2, 2 cuts the planes at points P and Q respectively

Length PQ = l

To find out:

[2l]

Where [] is greatest integer function

Solution:

Let the points that cut the planes be λ, 2λ, 2λ

Then

For plane 1

3(λ) - 2(2λ) + 6(2λ) + 12 = 0

⇒ 11λ = -12

⇒λ = -12/11

For plane 2

3(λ) - 2(2λ) + 6(2λ) - 2 = 0

⇒ 11λ = 2

⇒ λ = 2/11

Therefore, the coordinates of point P is (-12/11, -24/11, -24/11)

And the coordinates of point Q is (2/11, 4/11, 4/11)

$PQ=\sqrt{(\frac{2}{11}+\frac{12}{11})^2+(\frac{4}{11}+\frac{24}{11})^2+(\frac{4}{11}+\frac{24}{11})^2}$

$\implies PQ=\frac{1}{11}\sqrt{14^2+35^2+35^2}$

$\implies PQ=\frac{\sqrt{2646}}{11}$

$\implies PQ=4.67$

Thus, l = 4.68

2l = 2 × 4.68 = 9.36

Therefore,

[2l] = [9.36] = 9

Hope this answer is helpful.

Know More:

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