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NTA Abhyas JEE Main2020MathematicsThree Dimensional GeometryPractice

If the length of the projection of the line segment joining the points 1,2 , - 1 and 3 ,5 , 5 on the plane 3 x - 4 y + 12 z = 5 is equal to d units, then the value of 169 d 2 is equal to

Correct answer

3925

Step-by-step solution

Let, point A be 1,2 , - 1 , B be 3,5 , 5 A B → = 2 i ^ + 3 j ^ + 6 k ^ If θ is the angle between the normal vector to the plane and A B → then, the length of the projection = A B → sin ⁡ θ Normal vector = 3 i ^ - 4 j ^ + 12 k ^ So, cos ⁡ θ = A B → ⋅ n → A B → n → = 6 - 12 + 72 7 13 = 66 91 Length of the projection = d = 7 × 3925 91 = 5 157 13 units

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