Question

prove that the equation of straight line whose portion of the line intercepted between co-ordinate axis is bisected at that point (h,k) is $\frac{x}{2h}+\frac{x}{2k}=1$

Answer 1

Appropriate Question :-

Prove that the equation of straight line whose portion of the line intercepted between co-ordinate axis is bisected at that point (h,k) is $\dfrac{x}{2h}+\dfrac{y}{2k}=1$

$\large\underline{\sf{Solution-}}$

Let assume that required equation of line makes an intercept of a units on positive direction of x axis and intercept of b units on positive direction of y axis.

Let further assume that line meets the x - axis at A (a, 0) and meets the y - axis at B (0, b)

So, equation of line which makes an intercept of a and b units on x axis and y axis respectively is given by

$\rm \: \dfrac{x}{a} + \dfrac{y}{b} = 1 - - - (1)$

Now, further given that portion of the line (1) intercepted between co-ordinate axis is bisected at that point P (h,k).

Now, AB be the portion of line intercepted between the axis.

So, P (h, k) is the midpoint of A (a, 0) and B (0, b).

So, by using Midpoint Formula, we get

$\rm \: (h, \: k) \: = \: \bigg(\dfrac{a + 0}{2}, \: \dfrac{0 + b}{2} \bigg)$

$\rm \: (h, \: k) \: = \: \bigg(\dfrac{a}{2}, \: \dfrac{b}{2} \bigg)$

$\rm\implies \:a \: = \: 2h \: \: \: and \: \: \: b \: = \: 2k$

On substituting the values of a and b in equation (1), we get

$\rm\implies \:\rm \: \dfrac{x}{2h} + \dfrac{y}{2k} = 1$

Hence, Proved

$\rule{190pt}{2pt}$

Additional Information :-

Different forms of equations of a straight line

1. Equations of horizontal and vertical lines

Equation of line parallel to x - axis passes through the point (a, b) is y = b.

Equation of line parallel to y - axis passes through the point (a, b) is x = a.

2. Point-slope form equation of line

Equation of line passing through the point (a, b) having slope m is y - b = m(x - a)

3. Slope-intercept form equation of line

Equation of line which makes an intercept of c units on y axis and having slope m is y = mx + c.

4. Intercept Form of Line

Equation of line which makes an intercept of a and b units on x - axis and y - axis respectively is x/a + y/b = 1.

5. Normal form of Line

Equation of line which is at a distance of p units from the origin and perpendicular makes an angle β with the positive X-axis is x cosβ + y sinβ = p.