Misc 23 - The planes: 2x - y + 4z = 5, 5x - 2.5y + 10z = 6 are - Miscellaneous

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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise
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Misc 23 (Method 1) The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through 0,0, 5﷮4﷯﷯ Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos θ = 𝑨﷮𝟏﷯ 𝑨﷮𝟐﷯ + 𝑩﷮𝟏﷯ 𝑩﷮𝟐﷯ + 𝑪﷮𝟏﷯ 𝑪﷮𝟐﷯﷮ ﷮ 𝑨﷮𝟏﷯﷮𝟐﷯ + 𝑩﷮𝟏﷯﷮𝟐﷯ + 𝑪﷮𝟏﷯﷮𝟐﷯﷯ ﷮ 𝑨﷮𝟐﷯﷮𝟐﷯ + 𝑩﷮𝟐﷯﷮𝟐﷯ + 𝑪﷮𝟐﷯﷮𝟐﷯﷯﷯ Given the two planes are So, cos 𝜃 = 2 × 10﷯ + −1 × −5﷯ + (4 × 20)﷮ ﷮ 2﷮2﷯ + ( −1)﷮2﷯ + 4﷮2﷯﷯ ﷮ 10﷮2 ﷯+ ( −5)﷮2﷯ + 20﷮2﷯﷯﷯﷯ = 20 + 5 + 80﷮ ﷮4 + 1 + 16﷯ ﷮100 + 25 + 400﷯﷯﷯ = 105﷮ ﷮21﷯ ﷮525﷯﷯﷯ = 105﷮ ﷮21﷯ × ﷮25 × 21﷯﷯﷯ = 105﷮ ﷮21﷯ × 5 ﷮21﷯﷯﷯ = 105﷮21 × 5﷯﷯ = 1 So, cos θ = 1 ∴ θ = 0° Since angle between the planes is 0°, Therefore, the planes are parallel. So, option (B) is correct Misc 23 (Method 2) The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through 0,0, 5﷮4﷯﷯ Given, two planes are Two lines are parallel if their direction ratios are proportional. 𝐴﷮1﷯﷮ 𝐴﷮2﷯﷯ = 2﷮10﷯ = 1﷮5﷯ , 𝐵﷮1﷯﷮ 𝐵﷮2﷯﷯ = −1﷮−5﷯ = 1﷮5﷯ , 𝐶﷮1﷯﷮ 𝐶﷮2﷯﷯ = 4﷮20﷯ = 1﷮5﷯ Since, 𝑨﷮𝟏﷯﷮ 𝑨﷮𝟐﷯﷯ = 𝑩﷮𝟏﷯﷮ 𝑩﷮𝟐﷯﷯ = 𝑪﷮𝟏﷯﷮ 𝑪﷮𝟐﷯﷯ = 𝟏﷮𝟓﷯ , Therefore, the normal vectors of the two planes are parallel. So, the two planes are parallel. So, option (B) is correct

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