KCET · Maths · Three Dimensional Geometry
The three points \(A(2, 4, 3), B(4, a, 9)\) and \(C(10, -1, 7)\) form a right-angled triangle with \(\angle B = 90^\circ\), then the value of "a" is
- A \(1\) or \(4\)
- B \(-1\) or \(4\)
- C \(1\) or \(-4\)
- D \(-1\) or \(-4\)
Answer & Solution
Correct Answer
(B) \(-1\) or \(4\)
Step-by-step Solution
Detailed explanation
The given points are \(A(2, 4, 3)\), \(B(4, a, 9)\), and \(C(10, -1, 7)\).
The direction ratios of the line segment \(AB\) are \((2 - 4, 4 - a, 3 - 9) = (-2, 4 - a, -6)\).
The direction ratios of the line segment \(BC\) are \((10 - 4, -1 - a, 7 - 9) = (6, -1 - a, -2)\).
Since \(\angle B = 90^\circ\), the line segments \(AB\) and \(BC\) are perpendicular to each other. Therefore, the dot product of their direction vectors is zero.
\((-2)(6) + (4 - a)(-1 - a) + (-6)(-2) = 0\)
\(-12 - (4 - a)(1 + a) + 12 = 0\)
\((a - 4)(a + 1) = 0\)
\(a = 4\) or \(a = -1\)
Answer: \(-1\) or \(4\)
The direction ratios of the line segment \(AB\) are \((2 - 4, 4 - a, 3 - 9) = (-2, 4 - a, -6)\).
The direction ratios of the line segment \(BC\) are \((10 - 4, -1 - a, 7 - 9) = (6, -1 - a, -2)\).
Since \(\angle B = 90^\circ\), the line segments \(AB\) and \(BC\) are perpendicular to each other. Therefore, the dot product of their direction vectors is zero.
\((-2)(6) + (4 - a)(-1 - a) + (-6)(-2) = 0\)
\(-12 - (4 - a)(1 + a) + 12 = 0\)
\((a - 4)(a + 1) = 0\)
\(a = 4\) or \(a = -1\)
Answer: \(-1\) or \(4\)
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