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प्रश्न
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`
उत्तर
Let AB be the line segment between the axes and let P (a, b) be its mid-point.
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 and y/2 = b`
∴ x = 2a and y = 2a
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
i.e. bx + ay = 2ab
On dividing both sides by ab, we obtain
`(bx)/(ab) + (ay)/(ab) + (2ab)/(ab)`
= `x/a + y/b = 2`
Thus, the equation of the line is `x/a + y/b = 2`
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