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

Asked by Abhisek | 11 months ago |  66

##### Solution :-

Let AB be a line segment whose midpoint is P (a, b).

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

$$(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2})$$

$$(\dfrac{0+x}{2}, \dfrac{y+0}{2})$$ = (a,b)

$$(\dfrac{x}{2}, \dfrac{y}{2})$$ = (a,b)

a = $$\dfrac{x}{2}$$ and

b = $$\dfrac{y}{2}$$

x = 2a and y = 2b

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

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

$$y-2b=\dfrac{0-2b}{2a-0}(x-0)$$

$$y-2b=\dfrac{-2b}{2a}(x)$$

$$y-2b=\dfrac{-b}{a}(x)$$

a (y – 2b) = -bx

ay – 2ab = -bx

bx + ay = 2ab

Divide both the sides with ab, then

$$\dfrac{bx}{ab}+\dfrac{ay}{ab}=\dfrac{2ab}{ab}$$

$$\dfrac{x}{a}+\dfrac{y}{b}=2$$

Hence proved.

Answered by Pragya Singh | 11 months ago

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