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Question
If the distance of a point from the points (2, 1) and (1, 2) are in the ratio 2 :1, then find the locus of the point.
Solution
Let P(x1, y1) be any point on the locus.
Let A(2, 1) and B(1, 2) be the given point.
Given that PA : PB = 2 : 1
i.e. `"PA"/"PB" = 2/1`
PA = 2PB
PA2 = 4PB2
(x1 - 2)2 + (y1 - 1)2 = 4[(x1 - 1)2 + (y1 - 2)2]
`x_1^2 - 4x_1 + 4 + y_1^2 - 2y_1 + 1 = 4[x_1^2 - 2x_1 + 1 + y_1^2 - 4y_1 + 4]`
`x_1^2 + `y_1^2 - 4x_1 - 2y_1 + 5 = 4x_1^2 - 8x_1 + 4y_1^2 - 16y_1 + 20`
∴ `- 3x_1^2 - 3y_1^2 + 4x_1 + 14y_1 - 15 = 0`
∴ `3x_1^2 + 3y_1^2 - 4x_1 - 14y_1 + 15 = 0`
∴ The locus of the point (x1, y1) is 3x2 + 3y2 – 4x – 14y + 15 = 0
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