#geometry problem

Vectors and 3D Geometry

XY-plane divides the line joining the points A(2, 3, -5) and B(-1, -2, -3) in the ratio

Here's the idea in simple terms: we verify 5 ∶ 3 externally by full calculation: use standard identity, substitute, simplify step by step to reach 5 ∶ 3 externally.

Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry

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Vectors and 3D Geometry

If | a| = 3, | b | = 4 and | a - b | = 5, then what is the value of | a + b | ?

The logic is straightforward: using formula we get 5. Calculation gives this directly, check steps: apply rule, substitute values, simplify to get 5.

Ref: NCERT Mathematics Class XII — Vectors and 3D Geometry

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Coordinate Geometry

The area of triangle with vertices (K, 0), (4, 0), (0, 2) is 4 square units, then value of K is

Applying the basic concept, complete solution: write formula, plug data, do arithmetic. Final value matches 0 or 8. Simple and accurate.

Ref: NCERT Mathematics Class XI — Coordinate Geometry

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Matrices and Determinants 1

An equilateral triangle has each side equal to a. If the co-ordinates of its vertices are (x₁, y₁); (x₂, y₂): (x₃, y₃) t

We know that area =½|det| where det=|x₁ y₁ 1;x₂ y₂ 1;x₃ y₃ 1|. Equilateral area=√3/4 a². So ½|det|=√3/4 a², det²=3a⁴/4.

Ref: NCERT Mathematics Class XII — Matrices and Determinants

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