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

Solution:

Let A and B be the y and x intercepts of line that corresponds to the line segment whose midpoint is P (a, b).

Let the coordinates of A and B be (0, y) and (x, 0) respectively.

Since P (a, b) is the mid-point of AB,

[(0 + x)/2, (y + 0)/2] = (a, b)

[(x)/2, (y)/2] = (a, b)

x/2 = a, y/2 = b

⇒ x = 2a and y = 2b

Thus, the respective coordinates of A and B are (0, 2b) and (2a, 0).

The equation of the line passing through points (0, 2b) and (2a, 0) is

(y - 2b) = (0 - 2b)/(2a - 0) (x - 0)

(y - 2b) = - 2b/2a (x)

a ( y - 2b) = - bx

ay - 2ab = - bx

bx + ay = 2ab

On dividing both sides by ab , we obtain

bx/ab + ay/ab = 2ab/ab

⇒ x/a + y/b = 2

Hence, the equation of the line is x/a + y/b = 2

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NCERT Solutions Class 11 Maths Chapter 10 Exercise 10.2 Question 18

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

Summary:

The equation of the line that corresponds to the line segment whose midpoint is P (a, b) is x/a + y/b = 2