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Equation of a Line Perpendicular to a Given Line

Assertion (A)...

Question

Assertion (A)
If the points A(8, 1), B(k, −4) and C(2, −5) are collinear, then k = 4.

Reason (R)
Three points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are collinear only when x₁(y₂− y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) = 0.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

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Solution

The given points are $A (8, 1), B (k, - 4) and C (2, - 5)$
Here, $(x_{1} = 8, y_{1} = 1), (x_{2} = k, y_{2} = - 4) and (x_{3} = 2, y_{3} = - 5)$
Points A,B and C are collinear.
$\Rightarrow x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) = 0 \Rightarrow 8 (- 4 + 5) + k (- 5 - 1) + 2 (1 + 4) = 0 \Rightarrow 8 (1) + k (- 6) + 2 (5) = 0 \Rightarrow - 6 k = - 18 \Rightarrow k = 3$
Hence, Assertion(A) is false.
Reason(R):
The given statement is true.
Hence, the correct answer is (d).

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