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19/01/2022 4:36 pm
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Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x – y + z + 3 = 0 and let R (3, 5, γ) be a point of this plane. Then the square of the length of the line segment SR is __________.
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19/01/2022 4:47 pm
Since R (3, 5, γ) lies on the plane 2x – y + z + 3 = 0.
Therefore, 6 – 5 + γ + 3 = 0
⇒ γ = -4
Now,
dr's of line QS are 2, -1, 1
equation of line QS is
\(\frac{x-1}{2}\) = \(\frac{y-3}{-1}\) = \(\frac{z-4}{1}\) = λ
⇒ F(2λ + 1, -λ + 3, λ + 4)
F lies in the plane
⇒ 2(2λ + 1) – (–λ + 3) + (λ + 4) + 3 = 0
⇒ 4λ + 2 + λ – 3 + λ + 7 = 0
⇒ 6λ + 6 = 0 ⇒ λ = –1.
⇒ F(–1,4,3)
Since, F is mid-point of QS.
Therefore, coordinated of S are (-3, 5, 2).
So, SR = \(\sqrt{36 + 0 +36}\) = \(\sqrt{72}\)
SR2 = 72