Let $X,$ $Y,$ and $Z$ be points such that $\frac{XZ}{XY} = \frac{ZY}{XY} = \frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$?

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!!!