Question

The equation of the line which is such that the portion of line segment intercepted between the coordinate axes is bisected at $(4,-3)$, is

• A. $3x+4y=24$
• B. $3x–4y=12$
• C. $3x–4y=24$
• D. $4x–3y=24$

Answer 1

The correct option is C

$3x–4y=24$

Explanation of the correct option.

Step $1$: Find the intercept points.

Let the line intercepts the $x$ axis at $(a,0)$ and $y$axis at $(0,-b).$

Let's draw the figure.

Given: $(4,-3)$ is the midpoint of the line.

So by using mid point formula $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

$$\begin{gathered} \frac{a+0}{2}=4 \\ \Rightarrow a=8 \end{gathered}$$

$$\begin{gathered} \frac{0-b}{2}=-3 \\ \Rightarrow b=6 \end{gathered}$$

Step $2$: Find the slope of the line .

Since slope is given by $\mathbf{m}\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{2}}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}$

Slope of the line passing through $(8,0)$ and $(0,-6)$ is,

$$\begin{gathered} m=\frac{-6-0}{0-8} \\ \Rightarrow m=\frac{3}{4} \end{gathered}$$

Step $3$: Find the equation of the line .

Since line passing through point $\left(x_1,y_1\right)$ is given by $y–y_1=m(x–x_1)$

Required equation of line passing through $(8,0)$ is

$y–0=\frac{3}{4}(x–8)$

$\Rightarrow$$4y–3x+24=0$

$\Rightarrow$ $3x–4y=24$

Hence Option $\left(\mathrm{C}\right)$ is the correct answer.