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A plane passing through the points (0, −1, 0) and (0, 0, 1) and making an angle π/4 with the plane y − z + 5 = 0, also passes through the point:
- (−√2, 1, −4)
- (√2, −1, 4)
- (−√2, −1, −4)
- (√2, 1, 4)
Correct answer: (√2, 1, 4)
Solution
A plane through (0,-1,0) and (0,0,1) has the form a x + b y + c z = d with c = d = -b, normal (a, b, -b). The pi/4 condition with normal (0,1,-1) gives a^2 = 2 b^2, so the plane is sqrt(2) x + y - z = -1. Testing the options, (sqrt(2), 1, 4) gives 2 + 1 - 4 = -1, so it lies on the plane.
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