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.

Answer 1

The given points are $A\left(8,1\right),B\left(k,-4\right)\text{and}C\left(2,-5\right)$
Here, $\left(x_1=8,y_1=1\right),\left(x_2=k,y_2=-4\right)\text{and}\left(x_3=2,y_3=-5\right)$
Points A,B and C are collinear.
$$\begin{gathered} \Rightarrow x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow 8\left(-4+5\right)+k\left(-5-1\right)+2\left(1+4\right)=0 \\ \Rightarrow 8\left(1\right)+k\left(-6\right)+2\left(5\right)=0 \\ \Rightarrow -6k=-18 \\ \Rightarrow k=3 \end{gathered}$$
Hence, Assertion(A) is false.
Reason(R):
The given statement is true.
Hence, the correct answer is (d).