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The ratio in which the point (1/2,6) divides the line segment joining the points (3, 5) and  (–7, 9) is
  • a)
    3 : 1
  • b)
    1 : 3
  • c)
    1 : 2
  • d)
    2 : 1
Correct answer is option 'B'. Can you explain this answer?
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The ratio in which the point (1/2,6)divides the line segment joining t...
Let (1/2, 6) divide the line segment joining the points (3, 5) and (–7, 9) in the ratio of K: 1
Using section formula, the co–ordinate of the dividing point are given as

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The ratio in which the point (1/2,6)divides the line segment joining t...
Let's call the point (1/2, 6) as P, the point (3, 5) as A, and the point (x, y) as B.

To find the ratio in which P divides the line segment AB, we can use the formula for dividing a line segment in a given ratio:

(x, y) = ((rx2 + qx1)/(r + q), (ry2 + qy1)/(r + q))

where (x1, y1) and (x2, y2) are the coordinates of points A and B respectively, and r and q are the ratios in which point P divides the line segment AB.

Since point P divides the line segment AB, we have the following equations:

x = (rx2 + qx1)/(r + q) and y = (ry2 + qy1)/(r + q)

Substituting the given values, we have:

1/2 = (r*3 + q*1)/(r + q) and 6 = (r*5 + q*6)/(r + q)

Simplifying the first equation:

1/2 = (3r + q)/(r + q)
2 = 3r + q

Simplifying the second equation:

6 = 5r + 6q

We now have a system of two equations with two variables. Solving this system will give us the values of r and q, which represent the ratio in which point P divides the line segment AB.

Multiplying the first equation by 2 and subtracting it from the second equation:

6 - 2(2) = 5r + 6q - 2(3r + q)
6 - 4 = 5r + 6q - 6r - 2q
2 = -r + 4q

Dividing both sides of this equation by 2:

1 = -r/2 + 2q

Rearranging the equation:

r/2 = 2q - 1
r = 4q - 2

Substituting this value of r in the first equation:

1/2 = (3(4q - 2) + q)/(4q - 2 + q)
1/2 = (12q - 6 + q)/(5q - 2)
1/2 = (13q - 6)/(5q - 2)
2(13q - 6) = 1(5q - 2)
26q - 12 = 5q - 2
26q - 5q = 12 - 2
21q = 10
q = 10/21

Substituting this value of q in the equation r = 4q - 2:

r = 4(10/21) - 2
r = 40/21 - 2
r = (40 - 42)/21
r = -2/21

Therefore, the ratio in which the point (1/2, 6) divides the line segment joining the points (3, 5) and (x, y) is -2/21:10/21, or simply -2:10.
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