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प्रश्न
- Write down the co-ordinates of the point P that divides the line joining A(−4, 1) and B(17, 10) in the ratio 1 : 2.
- Calculate the distance OP, where O is the origin.
- In what ratio does the y-axis divide the line AB?
उत्तर
i. Co-ordinates of point P are
`((1 xx 17 + 2 xx (-4))/(1 + 2),(1 xx 10 + 2 xx 1)/(1 + 2))`
= `((17 - 8)/3, (10 + 2)/3)`
= `(9/3, 12/3)`
= (3, 4)
ii. `OP = sqrt((0 - 3)^2 + (0 - 4)^2)`
`OP = sqrt(9 + 16)`
`OP = sqrt(25)`
OP = 5 units
iii. Let AB be divided by the point P(0, y) lying on y-axis in the ratio k : 1
∴ `(0, y) = ((k xx 17 + 1 xx (-4))/(k + 1),(k xx 10 + 1 xx 1)/(k + 1))`
`=> (0, y) = ((17k - 4)/(k + 1),(10k + 1)/(k + 1))`
`=> 0 = (17k - 4)/(k + 1)`
`=> 17k - 4 = 0`
`=> k = 4/17`
Thus, the ratio in which the y-axis divide the line AB is 4 : 17.
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The points A(x1, y1), B(x2, y2) and C(x3, y3) are the vertices of ∆ABC. Find the coordinates of points Q and R on medians BE and CF, respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1
Complete the following activity to find the coordinates of point P which divides seg AB in the ratio 3:1 where A(4, – 3) and B(8, 5).
Activity:
∴ By section formula,
∴ x = `("m"x_2 + "n"x_1)/square`,
∴ x = `(3 xx 8 + 1 xx 4)/(3 + 1)`,
= `(square + 4)/4`,
∴ x = `square`,
∴ y = `square/("m" + "n")`
∴ y = `(3 xx 5 + 1 xx (-3))/(3 + 1)`
= `(square - 3)/4`
∴ y = `square`
