Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #877
KNOWLEDGE BASED
APPLY
2 Marks 2023 VSA
Find the vector and the cartesian equations of a line that passes through the point A(1,2,-1) and parallel to the line 5x-25=14-7y=35z.
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Step-by-Step Solution

  1. Rewrite the given equation of the line in standard form: 5x - 25 = 14 - 7y = 35z can be written as 5(x - 5) = -7(y - 2) = 35z. Dividing by 35, we get (x - 5)/7 = (y - 2)/(-5) = z/1.
  2. The direction ratios of the given line are 7, -5, and 1. Since the required line is parallel to the given line, its direction ratios will be the same.
  3. The required line passes through the point A(1, 2, -1). Therefore, a = 1, b = 2, and c = -1.
  4. The vector equation of the line is given by: r = a + λb, where a is the position vector of the point and b is the direction vector. Here, a = i + 2j - k and b = 7i - 5j + k. So, the vector equation is r = (i + 2j - k) + λ(7i - 5j + k).
  5. The cartesian equation of the line is given by: (x - a)/l = (y - b)/m = (z - c)/n, where (l, m, n) are the direction ratios. Here, a = 1, b = 2, c = -1, l = 7, m = -5, and n = 1. So, the cartesian equation is (x - 1)/7 = (y - 2)/(-5) = (z + 1)/1.

Correct Answer: Vector equation: r = (i + 2j - k) + λ(7i - 5j + k); Cartesian equation: (x - 1)/7 = (y - 2)/(-5) = (z + 1)/1<\/strong>

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires students to apply their knowledge of vector and cartesian equations of a line to find the equations given a point and a parallel line.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to derive the vector and cartesian equations of a line. This involves manipulating the given equations, identifying direction ratios, and applying the standard formulas.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of formulas and concepts related to 3D geometry as covered in the textbook.

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