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Mathematics

The point (x, y) is equidistant from the points (3, 6) and (-3, 4); the relation between x and y is :

  1. 3x - y = 5

  2. 3x - y - 5 = 0

  3. y - 3x = 0

  4. 3x + y = 5

Distance Formula

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Answer

Given (x, y) is equidistant from (3, 6) and (-3, 4)

i.e. distance between (x, y) and (3, 6) = distance between (x, y) and (-3, 4)

(x−3)2+(y−6)2=(x−(−3))2+(y−4)2⇒(x−3)2+(y−6)2=(x+3)2+(y−4)2⇒(x−3)2+(y−6)2=(x+3)2+(y−4)2⇒x2+9−6x+y2+36−12y=x2+9+6x+y2+16−8y⇒36−16=6x+6x+12y−8y⇒20=12x+4y⇒3x+y=5\sqrt{(x - 3)^2 + (y - 6)^2} = \sqrt{(x - (-3))^2 + (y - 4)^2}\\[1em] ⇒ \sqrt{(x - 3)^2 + (y - 6)^2} = \sqrt{(x + 3)^2 + (y - 4)^2}\\[1em] ⇒ (x - 3)^2 + (y - 6)^2 = (x + 3)^2 + (y - 4)^2\\[1em] ⇒ x^2 + 9 - 6x + y^2 + 36 - 12y = x^2 + 9 + 6x + y^2 + 16 -8y\\[1em] ⇒ 36 - 16 = 6x + 6x + 12y -8y\\[1em] ⇒ 20 = 12x + 4y\\[1em] ⇒ 3x + y = 5

Hence, option 4 is the correct option.

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