What is the equation of the line through (1, 2) so that the segment of the line intercepted between the axes is bisected at this point?

Concept:

Intercept form of line: \(\rm \frac {x}{a} + \rm \frac {y}{b} = 1\), Where, b = x-intercept and a = y-intercept

Mid-point formula: Let A(x₁, y₁) and B(x₂, y₂) be any two points on the X-Y plane. Suppose point C is the mid-point of the line segment AB, then the co-ordinates of point C is: \((\rm \frac {x_1 + x_2}{2} , \rm \frac {y_1 + y_2}{2})\)

Calculation:

Here, the line segment is bisected by the points (1, 2) i.e. they are the midpoint of the line segment passes through the points (0, a) and (b, 0) 

\(\therefore (1,2)=(\frac{b+0}{2},\frac{0+a}{2})\)

∴ \(\frac{b+0}{2}=1 ~and~ \frac{0+a}{2}=2\)

⇒ b = 2 i.e., x-intercept 

and a = 4 i.e., y-intercept 

∴ Equation of line \( \rm \frac {x}{a} + \rm \frac {y}{b} = 1\)

\( \rm \frac {x}{2} + \rm \frac {y}{4} = 1\)

⇒ 2x + y = 4

Hence, option (3) is correct.