Question 29 (A) - 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 (A) Find the distance of the point ( 2,−1, 3) from the line 𝑟 ⃗=(2ı ˆ−ȷ ˆ+2𝑘 ˆ )+𝜇(3ı ˆ+6ȷ ˆ+2𝑘 ˆ ) measured parallel to the z -axis.Blue line AB is parallel to z-axis Let Point A (2, –1, 3) Let point B the point on line 𝑟 ̂ such that AB is parallel to z-axis Equation of line is 𝒓 ⃗ = 2𝒊 ̂ – 𝒋 ̂ + 2𝒌 ̂ + 𝝁(3𝒊 ̂ + 6𝒋 ̂ + 2𝒌 ̂) We need to find Distance AB To find AB, we need to find point B Finding Point B Since point B lies on line 𝒓 ⃗ Now, 𝑟 ⃗ = 2𝑖 ̂ – 𝑗 ̂ + 2𝑘 ̂ + 𝜇(3𝑖 ̂ + 6𝑗 ̂ + 2𝑘 ̂) 𝑟 ⃗ = 2𝑖 ̂ – 𝑗 ̂ + 2𝑘 ̂ + 3𝜇𝑖 ̂ + 6𝜇𝑗 ̂ + 2𝜇𝑘 ̂ 𝒓 ⃗ = (2 + 3𝝁")" 𝒊 ̂ + (1 + 6𝝁")" 𝒋 ̂ + (2+2𝝁")" 𝒌 ̂ So, x = 2 + 𝟑𝝁 y = –1 + 𝟔𝝁 z = 2 + 𝟐𝝁 ∴ B = (3𝜇+2, 6𝜇−1, 2𝜇+2) 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, 3) B (3𝜇+2, 6𝜇−1, 2𝜇+2) Direction ratios of AB = 3𝜇+2−2, 6𝜇−1−(−1), 2𝜇+2−3 = 3𝜇, 6𝜇, 2𝜇−1 Since AB and z-axis are parallel The x and y component should be zero Equating x-component 3𝜇=0 𝝁=𝟎 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 = 2 + 3𝜇 = 2 + 3(0) = 2 y = –1 + 6𝜇 = –1 + 6(0) = –1 z = 2 + 2𝜇 = 2 + 2(0) = 2 ∴ B = (2, –1, 2) Thus, Distance between A(2, –1, 3) and B (2, –1, 2) AB = √((2−2)^2+((−1) −(−1))^2+(2−3)^2 ) = √(0^2+0^2+(−1)^2 ) = √1 = 1 unit Thus, required distance is 1 unit