Question 1139586
point p is at (3,-8)
point r is at (10,y)


the distance between them is 25.


point r is in the first quadrant.


this means that y has to be positive.


the distance between point p and point r is equal to sqrt((y+8)^2 + (10-3)^2).


simplify this to get distance between points p and r is equal to sqrt((y+8)^2 + 49)


since the distance between points p and r is 25, then the formula becomes:


25 = sqrt((y+8)^2 + 49)


square both sides of the equation to get 625 = (y+8)^2 + 49


simplify to get 625 = y^2 + 16y + 64 + 49


combine like terms to get 625 = y^2 + 16y + 113


subtract 625 from both sides of the equation to get 0 = y^2 + 16y - 512.


factor this quadratic equation to get (y + 32) * (y - 16) = 0


solve for y to get y = -32 or 16.


y is positive, so y has to be 16.


your solution is that y = 16.


this means that point p = (3,-8) and point r = (10,16)


the distance between points p and r is equal to sqrt((16+8)^2 + (10-3)^2).


that becomes equal to sqrt((24)^2 + 7^2) which becomes equal to sqrt(625) which becomes equal to 25.


that confirms that, when y = 16, the distance between p and r is 25.


the equation of the line between points p and r is y = 24/7 * x -128/7.


the graph of that equation is shown below.


it shows that the points (3,-8) and (10,16) are both on the line, as they sh ould be.


<img src = "http://theo.x10hosting.com/2019/042241.jpg" alt="###" >