Answer:
Coordinates of point X = (-3, -21)
The sum of the coordinates of X = - 24
Step-by-step explanation:
There are 3 points, X, Y and Z.
Let the coordinates of point X be (x, y)
Y = (1, 7)
Z = (-1, -7)
(XZ/XY) = (ZY/XY) = (1/2)
The distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate system is given as
d = √[(x₁ - x₂)² + (y₁ - y₂)²]
Therefore,
XZ = √[(x + 1)² + (y + 7)²]
XY = √[(x - 1)² + (y - 7)²]
ZY = √[(-1 - 1)² + (-7 - 7)²] = √(4 + 196) = 14.14
From the relation
(XZ/XY) = (ZY/XY) = (1/2)
We can deduce that
(XZ/XY) = (1/2)
(ZY/XY) = (1/2)
XZ = ZY
(XZ/XY) = (1/2)
XY = 2XZ
√[(x - 1)² + (y - 7)²] = 2(√[(x + 1)² + (y + 7)²]
Square both sides
(x - 1)² + (y - 7)² = 4[(x + 1)² + (y + 7)²]
x² - 2x + 1 + y² - 14y + 49 = 4[x² + 2x + 1 + y² + 14y + 49]
x² - 2x + 1 + y² - 14y + 49 = 4x² + 8x + 4 + 4y² + 56y + 196
3x² + 10x + 3 + 3y² + 70y + 147 = 0
3x² + 3y² + 10x + 70y + 150 = 0 (eqn 1)
(ZY/XY) = (1/2)
XY = 2ZY
√[(x - 1)² + (y - 7)²] = 2(14.142)
Square both sides
(x - 1)² + (y - 7)² = 4(200) = 800
x² - 2x + 1 + y² - 14y + 49 = 800
x² + y² - 2x - 14y - 750 = 0 (eqn 2)
XZ = ZY
√[(x + 1)² + (y + 7)²] = 14.142
Square both sides
(x + 1)² + (y + 7)² = 200
x² + 2x + 1 + y² + 14y + 49 = 200
x² + y² + 2x + 14y - 150 = 0 (eqn 3)
Rewriting the 3 equations together,
3x² + 3y² + 10x + 70y + 150 = 0
x² + y² - 2x - 14y - 750 = 0
x² + y² + 2x + 14y - 150 = 0
Make (x² + y²) the subject of formula in equation 3
(x² + y²) = -2x - 14y + 150
Substituting this into eqn 1
3x² + 3y² + 10x + 70y + 150 = 0
3(x² + y²) + 10x + 70y + 150 = 0
(x² + y²) = -2x - 14y + 150
3(-2x - 14y + 150) + 10x + 70y + 150 = 0
-6x - 42y + 450 + 10x + 70y + 150 = 0
4x + 28y + 600 = 0
x + 7y = -150
Substitute this into this expression (x² + y²) = -2x - 14y + 150
x = -7y - 150
(x² + y²) = -2x - 14y + 150
(-7y - 150)² + y² = -2(-7y - 150) - 14y + 150
49y² + 2100y + 22500 + y² = 14y + 300 - 14y + 150
50y² + 2100y + 22050 = 0
y² + 42y + 441 = 0
y = -21
x = -7y - 150 = -7(-21) - 150 = 147 - 150 = -3
Point X's coordinate = (x, y) = (-3, -21).
The sum of X's coordinate = -2 - 21 = -24
Hope this Helps!!!
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