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प्रश्न
ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.
उत्तर
Given ABCD is a square.
∴ AB = BC (all sides of a square are equal)
`sqrt (("x" - 5)^2 + ("y" - 4)^2) = sqrt (("x" + 1)^2 + ("y" - 6)^2)`
squaring both sides ,
x2 + 25 - 10x + y2 + 16 - 8y = x2 + 1 + 2x + y2 + 36 - 12y
⇒ - 12 x + 4y + 4 = 0
⇒ - 3x + y + 1 =0
y = 3x - 1 ......(1)
Also , each angle in a square measures 90°
By pythogoras theorem ,
AB2 + BC2 = AC2
⇒ (5 - x)2 + (4 - y)2 + (x + 1)2 + (y - 6)2 = 36 +4
⇒ 25 + x2 - 10x + 16 + y2 - 8y + x2 + 1 + 2x + y2 + 36 - 12y
⇒ 2x2 + 2y2 - 8x - 20y + 38 = 0
⇒ x2 + y2 - 4x - 10y + 19 = 0
⇒ x2 + (3x - 1)2 - 4x - 10(3x - 1)+ 19 = 0
⇒ x2 + 9x2 + 1 - 6x - 4x - 30x + 10 10 = 0
⇒ 10x2 - 40x + 30 = 0
⇒ x2 - 4x + 3 = 0
⇒ x2 - 3x - x + 3 = 0
⇒ x (x - 3) -1 (x - 3) = 0
⇒ (x - 1)(x - 3) = 0
x = 1 ,3
When , x = 1 , y = 3(1)-1 = 2 (1 , 2)
x = 3 , y = 3(3) - 1 = 8 (3 , 8)
Thus , coordinates of B and D are (1 , 2) and (3 , 8)
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