Find the point equidistant from given points

Question

For some reason I'm having a challenging time trying to figure out how to go about this problem. Do I use the midpoint formula or the distance formula maybe? Any help would be so appreciated :)

 

Find the point (0,b) on the y-axis that is equidistant from the points (4,4) and (5,−4).

Answer 1

This is should be a distance formula problem.  (Good that you asked the question.)

 

√[(4-0)²+(4-b)²] = √[(5-0)²+(-4-b)²]     is how you set up the equality of the two distance formulas.

 

√(16 + 16 - 8b + b²) = √(25 +16 + 8b + b²)        now square both sides and rewrite in standard form

 

b² - 8b + 32 = b² + 8b + 41                  isolate the b (notice the b² terms drop out)

 

-9 = 16b                                              solve for  b

 

-9/16 = b

 

You can check the number by substituting it for b in the above equality

√[(4-0)² + (4 + 9/16)^2] = √[(5-0)² + (-4 + 9/16)²]

 

 

 

 

Answer 2

A point on the y-axis can be expressed as (0,y), for some real number, y.

 

So, by the distance formula, √[(0-4)²+(y-4)²] = √[(0-5)²+(y-(-4))²] 

 

                                        √(y²-8y+32) = √(y²+8y+41)

 

Square both sides to get y²-8y+32 = y²+8y+41

 

                                      -8y+32 = 8y+41

 

                                           -16y = 9

 

                                               y = -9/16