Question

Assertion (A)
The point P(−1, 6) divides the join of A(−3, 10) and B(6, −8) in the ratio 2 : 7.

Reason (R)
If the point C(x, y) divides the join of A(x₁, y₁) and B(x₂, y₂) in the ratio m : n, then $x=\frac{m{x}_2+n{x}_1}{m+n}\mathrm{and}y=\frac{m{y}_2+n{y}_1}{m+n}.$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) 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 correct answer is (a)
Let the required ratio be k : 1.
Then, by section formula, the coordinates of C are
$P\left(\frac{6k-3}{k+1},\frac{-8k+10}{k+1}\right)$
Therefore,
$$\begin{gathered} \frac{6k-3}{k+1}=-1\text{and}\frac{-8k+10}{k+1}=6\left[\text{Since}P\left(-1,6\right)\text{is given}\right] \\ \Rightarrow 6k-3=-k-1\text{and}-8k+10=6k+6 \end{gathered}$$

$\Rightarrow 7k=2\text{and}-14k=-4$
$\Rightarrow k=\frac{2}{7}\text{in each case.}$
Therefore, the required ratio is $\frac{2}{7}:1$, which is same as 2 : 7.
Hence, Assertion(A) is true.
Reason(R):
The given statement is true.
Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).