Question

The equation of straight line equally inclined to the axes and equidistant from the points $(1,-2)$ and $(3,4)$ is $ax+by+c=0$ where:

• A. $a=1,b=1,c=1$
• B. $a=1,b=-1,c=-1$
• C. $a=1,b=1,c=2$
• D. None of these

Answer 1

The correct option is B

$a=1,b=-1,c=-1$

Determine the equation of the line :

The required line is equally inclined to the axes. It means that the slope of the line is $=\tan 45°=1$

It will make an equal angle with $x-axis$ and $y-axis$.

The required line is equidistant from $(1,-2)$ and $(3,4)$ It means that the required line passes through the midpoint of the line joining the two given points.

midpoint of the given points:

$$\begin{gathered} \Rightarrow \left(\frac{1+3}{2},\frac{-2+4}{2}\right) \\ \Rightarrow \left(2,1\right) \end{gathered}$$

Therefore, the line passes through the point $(2,1)$ and has slope $=1$.

Now, the equation of the required line is:

$$\begin{gathered} y-y_1=m(x-x_1) \\ \Rightarrow y-1=1(x-2) \\ \Rightarrow y-1=x-2 \\ \Rightarrow x-y-1=0 \end{gathered}$$

Comparing this equation $x-y-1=0$ with $ax+by+c=0$ we get:

$a=1,b=-1,c=-1$

Hence, the correct answer is Option (B).