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प्रश्न
Show that the lines x = O and y = O trisect the line segment formed by joining the points (-10, -4) and (5, 8). Find the points of trisection.
उत्तर
Let P (x, 0) lies on the line y = 0 i.e. x-axis and divides the line segment AB in the ratio k: 1.
Coordinates of P are,
P (x , 0) = P `((5"k" - 10)/("k" + 1) , (8"k" - 4)/("k" + 1))`
`=> 0 = (8"k" - 4)/("k" + 1) , (5"k" - 10)/("k" + 1) = "x"`
`=> 8"k" = 4 , (5(1/2) - 10)/(1/2 + "x") = "x"` ........from (1)
`=> "k" = 1/2` ........(1) , x = -5
Hence P(-5, 0) divides AB in the ratio 1: 2.
Let Q (0, y) lies on the line x = 0 i.e. y - axis and
divides the line segment AB in the ratio k1 : 1.
Coordinates of Q are
Q (0 , y) = Q `((5"k"_1 - 10)/("k"_1 + 1) , (8"k"_1 - 4)/("k"_1 + 1))`
`0 = (5"k"_1 - 10)/("k"_1 + 1) , "y" = (8"k"_1 - 4)/("k"_1 + 1)`
`=> 5 "k"_1 = 10 , "y" = (8(2) - 4)/(2 + 1)` .......from (2)
⇒ k1 = 2 .... (2) y = 4
Hence, Q(O, 4) divides in the ratio 2 : 1.
Hence proved Paid Qare the points of trisection of AB.
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