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Question
In what ratio is the line segment joining A(2, -3) and B(5, 6) divide by the x-axis? Also, find the coordinates of the pint of division.
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Solution
Let AB be divided by the x-axis in the ratio k : 1 at the point P.
Then, by section formula the coordination of P are
`p = ((5k+2)/(k+1),(6k-3)/(k+1))`
But P lies on the x-axis; so, its ordinate is 0.
Therefore , `(6k-3)/(k+1) = 0`
`⇒ 6k -3=0 ⇒ 6k =3 ⇒k = 3/6 ⇒ k = 1/2`
Therefore, the required ratio is `1/2:1 `, which is same as 1 : 2
Thus, the x-axis divides the line AB li the ratio 1 : 2 at the point P.
Applying `k=1/2` we get the coordinates of point.
`p((5k+1)/(k+1) , 0)`
`=p((5xx1/2+2)/(1/2+1),0)`
`= p (((5+4)/2)/((5+2)/2),0)`
`= p (9/3,0)`
= p (3,0)
Hence, the point of intersection of AB and the x-axis is P( 3,0).
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