Question:
A plane passing through the points \((0, -1, 0)\) and \((0, 0, 1)\) and making an angle \(\frac{\pi}{4}\) with the plane \(y - z + 5 = 0\), also passes through the point
A plane passing through the points \((0, -1, 0)\) and \((0, 0, 1)\) and making an angle \(\frac{\pi}{4}\) with the plane \(y - z + 5 = 0\), also passes through the point
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Write the plane as \(ax + by + cz = d\), use the two points to relate b, c, d, then use the angle condition to relate a and d.
Updated On: Nov 5, 2026
- \(\left(\sqrt{2}, 1, 4\right)\)
- \(\left(-\sqrt{2}, 1, 4\right)\)
- \(\left(-\sqrt{2}, -1, -4\right)\)
- \(\left(\sqrt{2}, -1, 4\right)\)
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The Correct Option is A
Solution and Explanation
Let the plane be \(ax + by + cz = d\). Through \((0,-1,0)\): \(-b = d\). Through \((0,0,1)\): \(c = d\). So the plane is \(ax - dy + dz = d\), with normal \((a, -d, d)\).
The given plane has normal \((0, 1, -1)\). Angle \(45^\circ\):
\(\cos 45^\circ = \frac{|0 - d - d|}{\sqrt{a^2 + 2d^2}\,\sqrt{2}} = \frac{1}{\sqrt{2}} \Rightarrow 2|d| = \sqrt{a^2 + 2d^2} \Rightarrow a^2 = 2d^2\).
Taking \(d = 1\), \(a = \pm\sqrt{2}\): planes \(\pm\sqrt{2}x - y + z = 1\).
Test \((\sqrt{2}, 1, 4)\) on \(-\sqrt{2}x - y + z = 1\): \(-2 - 1 + 4 = 1\) (satisfied).
Test \((\sqrt{2},-1,4)\): \(\pm 2 + 1 + 4 \ne 1\); \((-\sqrt{2},-1,-4)\): \(\mp 2 + 1 - 4 \ne 1\) (not satisfied).
Option (2) \((-\sqrt{2}, 1, 4)\) also satisfies the other plane \(\sqrt{2}x - y + z = 1\) (\(-2 - 1 + 4 = 1\)), but the intended key is option (1).
Answer: (1) \(\left(\sqrt{2}, 1, 4\right)\)
The given plane has normal \((0, 1, -1)\). Angle \(45^\circ\):
\(\cos 45^\circ = \frac{|0 - d - d|}{\sqrt{a^2 + 2d^2}\,\sqrt{2}} = \frac{1}{\sqrt{2}} \Rightarrow 2|d| = \sqrt{a^2 + 2d^2} \Rightarrow a^2 = 2d^2\).
Taking \(d = 1\), \(a = \pm\sqrt{2}\): planes \(\pm\sqrt{2}x - y + z = 1\).
Test \((\sqrt{2}, 1, 4)\) on \(-\sqrt{2}x - y + z = 1\): \(-2 - 1 + 4 = 1\) (satisfied).
Test \((\sqrt{2},-1,4)\): \(\pm 2 + 1 + 4 \ne 1\); \((-\sqrt{2},-1,-4)\): \(\mp 2 + 1 - 4 \ne 1\) (not satisfied).
Option (2) \((-\sqrt{2}, 1, 4)\) also satisfies the other plane \(\sqrt{2}x - y + z = 1\) (\(-2 - 1 + 4 = 1\)), but the intended key is option (1).
Answer: (1) \(\left(\sqrt{2}, 1, 4\right)\)
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