Respuesta :
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!!!
The sum of the X coordinates is -24
The given parameters are:
[tex]\frac{XZ}{XY} = \frac{ZY}{XY} = \frac{1}{2}[/tex]
[tex]Y = (1, 7)[/tex]
[tex]Z = (-1, -7)[/tex]
Start by calculating the distance YZ using the following distance formula
[tex]YZ = \sqrt{(x_2 -x_1)^2 + (y_2 -y_1)^2}[/tex]
This gives
[tex]YZ = \sqrt{(1--1)^2 + (7--7)^2}[/tex]
[tex]YZ = \sqrt{200}[/tex]
Simplify the root
[tex]YZ = 10\sqrt{2}[/tex]
[tex]\frac{XZ}{XY} = \frac{ZY}{XY} = \frac{1}{2}[/tex] means that:
[tex]XZ = ZY[/tex]
[tex]XY= 2XZ = 2ZY[/tex]
Calculate the distance XY
[tex]XY = \sqrt{(x-1)^2 + (y-7)^2}[/tex]
Calculate distance XZ
[tex]XZ = \sqrt{(x+1)^2 + (y+7)^2}[/tex]
So, we have:
[tex]\sqrt{(x-1)^2 + (y-7)^2} = 2 \times \sqrt{(x+1)^2 + (y+7)^2}[/tex]
Square both sides of the equation
[tex](x-1)^2 + (y-7)^2 = 4 \times [(x+1)^2 + (y+7)^2][/tex]
Expand
[tex]x\² - 2x + 1 + y\² - 14y + 49 = 4(x\² + 2x + 1 + y\² + 14y + 49)[/tex]
Open bracket
[tex]x\² - 2x + 1 + y\² - 14y + 49 = 4x\² + 8x + 4 + 4y\² + 56y + 196[/tex]
Collect like terms
[tex]4x^2 - x^2 + 8x + 2x + 4 - 1 + 4y^2 - y^2 + 56y + 14y + 196- 49 = 0[/tex]
Evaluate the like terms
[tex]3x\² + 3y\² + 10x + 70y + 150 = 0[/tex]
Recall that:
[tex]XY= 2XZ = 2ZY[/tex]
[tex]XY = \sqrt{(x-1)^2 + (y-7)^2}[/tex]
[tex]YZ = 10\sqrt{2}[/tex]
So, we have:
[tex]\sqrt{(x - 1)\² + (y - 7)\²} =2 \times 10\sqrt 2[/tex]
[tex]\sqrt{(x - 1)\² + (y - 7)\²} =20\sqrt 2[/tex]
Square both sides
[tex](x - 1)\² + (y - 7)\² = 800[/tex]
Expand
[tex]x\² - 2x + 1 + y\² - 14y + 49 = 800[/tex]
Collect like terms
[tex]x\² + y\² - 2x - 14y + 1 + 49 - 800 =0[/tex]
[tex]x\² + y\² - 2x - 14y - 750 = 0[/tex]
Also, we have:
[tex]XZ = ZY[/tex]
So, the equation becomes
[tex]\sqrt{(x + 1)\² + (y + 7)\²} = 10\sqrt 2[/tex]
Square both sides
[tex](x + 1)\² + (y + 7)\² = 200[/tex]
Expand
[tex]x\² + 2x + 1 + y\² + 14y + 49 = 200[/tex]
Collect like terms
[tex]x\² + y\² + 2x + 14y - 150 = 0[/tex]
So, we have:
[tex]3x\² + 3y\² + 10x + 70y + 150 = 0[/tex]
[tex]x\² + y\² - 2x - 14y - 750 = 0[/tex]
[tex]x\² + y\² + 2x + 14y - 150 = 0[/tex]
Using a graphing calculator, we have:
[tex](x, y) = (-3, -21)[/tex]
The above represents the X coordinates.
Add the coordinates:
[tex]X = -3 - 21[/tex]
[tex]X = - 24[/tex]
Hence, the sum of the X coordinates is -24
Read more about distance at:
https://brainly.com/question/17206319
