Question 29 (B) - CBSE Class 12 Sample Paper for 2026 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Question 29 (B) Find the point of intersection of the line 𝑟 ⃗=(3ı ˆ+𝑘 ˆ)+𝜇(ı ˆ+ȷ ˆ+𝑘 ˆ) and the line through (2,−1, 1) parallel to the z -axis. How far is this point from the z -axis?Blue line AB is parallel to z-axis And point B is line 𝒓 ⃗ Let Point A (2, –1, 1) Let point B the point on line 𝑟 ̂ such that AB is parallel to z-axis Equation of line is 𝒓 ⃗ = (3𝒊 ̂ + 𝒌 ̂) + 𝝁(𝒊 ̂ + 𝒋 ̂ + 𝒌 ̂) Now, point B is the point of intersection of line line 𝑟 ⃗=(3ı ˆ+𝑘 ˆ)+𝜇(ı ˆ+ȷ ˆ+𝑘 ˆ) and the line through (2,−1, 1) parallel to the z -axis Finding Point B Since point B lies on line 𝒓 ̂ Now, 𝑟 ⃗ = 3𝑖 ̂ + 𝑘 ̂ + 𝜇(𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂) 𝑟 ⃗ = 3𝑖 ̂ + 𝑘 ̂ + 𝜇𝑖 ̂ + 𝜇𝑗 ̂ + 𝜇𝑘 ̂ 𝒓 ⃗ = (3 + 𝝁")" 𝒊 ̂ + 𝝁𝒋 ̂ + (1 + 𝝁")" 𝒌 ̂ So, x = 3 + 𝝁 y = 𝝁 z = 1 + 𝝁 ∴ B = (3 + 𝜇, 𝜇,1 + 𝜇) Since AB is parallel to z-axis Direction cosines of z-axis are a = cos 90° , b = cos 90° , c = cos 0° a = 0 , b = 0, c = 1 a = 0 , b = 0, c = 1 ∴ Direction ratios of z – axis are 0, 0, 1 Note: Direction cosines and direction ratios of z-axis are same. We use Direction ratios here because finding Direction ratios of AB is easier. Direction ratio of AB For A (2, –1, 1) B (3 + 𝜇, 𝜇,1 + 𝜇) Direction ratios of AB = 3+𝜇−2 , 𝜇−(−1), 1+𝜇−1 = 𝟏+𝝁, 𝝁+𝟏, 𝝁 Since AB and z-axis are parallel The x and y component should be zero Equating x-component 𝜇+1=0 𝝁=−𝟏 Note: If lines are parallel, the direction ratios are proportional. Since here x and y components are zero, we directly make them equal. z-component cannot be made equal (it should be proportional) Thus, point B becomes x = 3 + 𝜇 = 3 – 1 = 2 y = 𝜇 = –1 z = 1 + 𝜇 = 1 + (–1) = 0 ∴ B = (2, –1, 0) Thus, required point B is (2, –1, 0) Now, we need to find Distance between B (2, –1, 2) and z-axis Distance of B (x, y, z) and z-axis = √(𝒙^𝟐+𝒚^𝟐 ) = √(2^2+〖(−1) 〗^2 ) = √(4+1) = √𝟓 units Thus, required distance is √𝟓 units