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Question
The point (x, y) is equidistant from the points (3, 4) and (−5, 6). Find a relation between x and y
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Solution
Let the point O be (x, y), A be (3, 4) and B be (−5, 6).
Distance = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
Given, OA = OB
`sqrt((x - 3)^2 + (y - 4)^2) = sqrt((x + 5)^2 + (y - 6)^2`
Squaring on both sides
(x – 3)2 + (y – 4)2 = (x + 5)2 + (y – 6)2
x2 – 6x + 9 + y2 – 8y + 16 = x2 + 10x + 25 + y2 – 12y + 36
x2 + y2 – 6x – 8y + 25 = x2 + y2 + 10x – 12y + 61
6x – 10x – 8y + 12y = 61 – 25
⇒ – 16x + 4y = 36 ........(÷ 4)
⇒ – 4x + y = 9
∴ The relation between x and y is y = 4x + 9
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