Question

A line meets the x-axis and y-axis at points $AandB$ respectively. If the middle point of $ABbe(x_{1_{}},y_{1_{}})$, then the equation of the line is

• A. $y_{1}x+x_{1}y=2x_1{y}_1$
• B. $x_{1}x+y_{1}y=2x_1{y}_1$
• C. $y_{1}x+x_{1}y=x_1{y}_1$
• D. $x_{1}x+y_{1}y=x_1{y}_1$

Answer 1

The correct option is A

$y_{1}x+x_{1}y=2x_1{y}_1$

Given the middle point of $ABis(x_1,y_1)$

Let the line meets the x-axis at $A$, y coordinate will be 0.

So $A=(2x_1,0)(\sin ce(x_1,y_1)isthemidpoint)$

The explanation for the correct option:

Let the line meets the y-axis at B, the x coordinate will be 0.

So $B=(0,2y_1)$

$\begin{array}{rcl}SlopeofAB& =& \frac{y_2–y_1}{x_2–x_1}\\ & =& \frac{0–2y_1}{2x_1–0}\\ & =& -\frac{y_1}{x_1}\end{array}$

Equation of line AB with slope $\begin{array}{rcl}m& =& -\frac{y_1}{x_1}\end{array}$ is

$\begin{array}{rcl}(y–y_1)& =& m(x–x_1)\\ (y–0)& =& -\frac{y_1}{x_1}(x–2x_1)\\ x_{1}y+x{y}_1& =& 2x_1{y}_1\\ y_{1}x+x_{1}y& =& 2x_1{y}_1\end{array}$

Hence option A is the answer.