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The locus of a point which divides a line segment AB = 4cm in 1 : 2, where A lies on the line y = x and B lies on the y = 2x is

The locus of a point which divides a line segment AB = 4cm in 1 : 2, where A lies on the line y = x and B lies on the y = 2x is

Grade:11

1 Answers

Vikas TU
14149 Points
3 years ago
Dear Student,
AB2 = (x1 – x2)2 + (x1 – 2x2)2 =  2x12 - 6 x1 x2 +  5x22 = 16.
Let the point which divides AB in the ratio 1:2 be X(x,y).
We have  XA2 = (x – x1)2 + (y – x1)2 = x2 + y2 + 2x12 - 2 x1 (x + y) = (4x1/3)2 = 16/9.
Similarly, XB2 = (x – x2)2 + (y – 2x2)2 = x2 + y2 + 5x22 - 2 x2 (x + 2y) = (4x2/3)2 = 64/9.
From the first equation, we have  x2 + y2 = 16/9 - 2x12 + 2 x1 (x + y).
Substituting this in the next equation, we get 16/9 - 2x12 + 2 x1 (x + y) + 5x22 - 2 x2 (x + 2y) = 64/9.
Simplifying this, we have 2(x1 – x2)x -  2(x1 + 2x2)y + 2x12 + 5x22 = 48/9 which is the locus.
Cheers!!
Regards,
Vikas (B. Tech. 4th year
Thapar University)

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