Question

If the points \((x_1, y_1),(x_2, y_2)\) and \((x _1 + x_2, y_1 + y_2) \) are collinear, then \(x_1y_2\) is equal to

(A) \(x_2y_1\)

(B) \(x_1y_1\)

(C) \(x_2y_2\)

(D) \(x_1x_2\)

Answer 1

Correct option is (A) \(x_2y_1\)

Method 1: 

When the points \(\left({x}_{1}, {y}_{{1}}\right),\left({x}_{2}, {y}_{2}\right)\) and \(\left({x}_{1}+{x}_{2}, {y}_{1}+{y}_{2}\right)\) are collinear in the Cartesian plane then

\(\left|\begin{array}{cc}{x}_{1}-{x}_{2} & {y}_{1}-{y}_{2} \\ {x}_{1}-\left({x}_{1}+{x}_{2}\right) & {y}_{1}-\left({y}_{1}+{y}_{2}\right)\end{array}\right|={0} \)

\(\Rightarrow\left|\begin{array}{cc}{x}_{1}-{x}_{2} & {y}_{1}-{y}_{2}\\ -{x}_{2} & -{y}_{2}\end{array}\right|=\left(-{x}_{1} {y}_{2}+{x}_{2} {y}_{2}+{x}_{2} {y}_{1}-{x}_{2} {y}_{2}\right)={0}\)

\(\Rightarrow {x}_{2} {y}_{1}={x}_{1} {y}_{2}\)

Method 2:

When the points \(\left({x}_{1}, {y}_{1}\right),\left({x}_{2}, {y}_{2}\right)\) and \(\left({x}_{1}+{x}_{2}, {y}_{1}+{y}_{2}\right)\) are collinear in the Cartesian plane then

\(\left|\begin{array}{ccc}{x}_{1} & {y}_{1} & {1} \\ {x}_{2} & {y}_{2} & {1} \\ {x}_{1}+{x}_{2} & {y}_{1}+{y}_{2} & 1\end{array}\right|=0\)

\(\Rightarrow {1} \cdot\left({x}_{2} {y}_{1}+{x}_{2} {y}_{2}-{x}_{1} {y}_{2}-{x}_{2} {y}_{2}\right)-{1}\left({x}_{1} {y}_{1}+{x}_{1} {y}_{2}-{x}_{1} {y}_{1}-{x}_{2} {y}_{1}\right)+\left({x}_{1} {y}_{2}-{x}_{2} {y}_{1}\right)={0}\)

\(\Rightarrow {x}_{2} {y}_{1}={x}_{1} {y}_{2}\)