The coordinate p = (a,0 is at an equal distance from origo as the coordinate Q = (3, 4). Solve algebraically possible values for a.

Question

That is my question, i think i need to use pythagorean theorem? The distance formula? This is a question on linear functions.

Answer 1

From geometry it is known that the locus of points equidistant from two specified points, P,Q,is  the line which is the  perpendicular  bisector of the line segment joining P and Q.  

 

In this problem  P = (0,0) and Q = (3,4).    The slope of the line segment joining P and Q is 4/3.   The slope of a line perpendicular to this is   -3/4.   The midpoint of the line segment  PQ is   (3/2, 2).      Therefore, the equation of the perpendicular bisector line is given by the point slope formula as:

   y- 2 = (-3/4) (x - 3/2)   or y = (-3/4) x + 25/8  

 

  From this equation, for y to be zero,     a must be   25/6 . 

Answer 2

First, get the distance from (0,0) to (3,4):

 

d = sqrt[(3-0)^2 + (4-0)^2] = sqrt(9+16) = 5 

 

Then set the distance from (0,0) to (a,0) equal to the distance from (0,0) to (3,4):

 

5 = sqrt[(a-0)^2 + (0-0)^2] = ±a

 

Then a = ±5

 

Now test it:

 

Distance from (5,0) to (0,0) is 5.  So is distance from (-5,0) to (0,0).