Question

Question 18 P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is x/a + y/b = 2

Class X1 - Maths -Straight Lines Page 220

Answer 1

Let equation of line is
x/h + y/k = 1----(1)
here, h and k are intercepts made by the line on x-axis and y-axis respectively.

now,
A/c to question,
P(a, b) is the midpoint of (h,0) and (0,k) .
Use formula,
Midpoint of (x1, y1) and (x2, y2) is {(x1 + x2)/2 , (y1 + y2)/2 }
So, a = ( 0 + h)/2
a = h/2 => h = 2a

b = (k + 0)/2
= k/2 => k = 2b

now, put this values in equation (1),
x/(2a) + y/(2b) = 1
x/a + y/b = 2

Hence proved

Answer 2

$\mathtt{ \huge{\fbox{SOLUTION \: : }}}$

Let , Q(0,k) and R(h,0) be the point whose mid point P (a, b)

This implies ,

x intercept is h and y intercept is k

We know that , the mid point between two points is given by

$\star \: \: \sf \large \fbox{x = \frac{x_{2} + x_{1}}{2} \: \: , \: \: y = \frac{ y _{2} + y_{1}}{2} }$

Thus,

$\sf \mapsto a= \frac{h + 0}{2} \: \: , \: \: b = \frac{0 + k}{2} \\ \\\sf \mapsto h = 2a \: \: , \: \: k = 2a$

We know that , the point intercept form is given by

$\star \: \: \sf \large\fbox{ \frac{x}{a} + \frac{y}{b} = 1}$

Where ,

b = x intercept

a = y intercept

So ,

$\sf \mapsto \frac{x}{2a} + \frac{y}{2b} = 1 \\ \\\sf \mapsto \frac{x}{a} + \frac{y}{b} = 2$

Hence , proved