Find all points on the x-axis that are a distance 8 from P(5, -7)

Question

smaller x value and larger x value

 

Answer 1

Another way to work this problem:

 

The points must be on a circle of radius 8 centered at (5,-7).  The equation of this circle is

 

(x-5)² + (y+7)² = 8² = 64

 

Since we want points on the x-axis, set y = 0 and get

 

(x-5)² + 49 = 64

 

(x-5)² = 15

 

x-5 = ±√15 

 

x = 5±√15 ≅ 8.873 or 1.127

 

Since y=0, the points are (8.873, 0) and (1.127, 0)

 

(Same answers, just a different way to work the problem)

 

 

 

Answer 2

You will use the distance formula:

 

d² = (x₁ - x₂)² + (y₁ - y₂)²

 

where:

d = 8

P = (x₁ , y₂) = (5, -7)

x₂ = any value

y₂ = 0

 

What we want to do is solve for x₂ from this formula.

 

(x₁ - x₂)² = d² - (y₁ - y₂)²

 

x₁ - x₂ = ±√[d² - (y₁ - y₂)²]

 

 

x₁ - x₂ = √[d² - (y₁ - y₂)²]                and             x₁ - x₂ = -√[d² - (y₁ - y₂)²]

 

Of these conjugates, you will find your two x values.

 

 

 

Answer 3

A point on the x-axis can be expressed as (x,0).

 

So, we want the distance from (x,0) to (5,-7) to be 8.

 

By the distance formula, we have √[(x-5)²+(0-(-7))²] = 8

 

Simplify to get √(x²-10x+74) = 8

 

Square both sides to obtain x²-10x+10 = 0

 

By the quadratic formula, x = (10 ±√60)/2

 

                                         = (10±2√15)/2 = 5±√15 ≈ 8.87 or 1.13

 

The points are approximately (8.87,0) and (1.13,0).