Final answer:
To find the constant a so that point P is equidistant from points A and B, set up and solve an equation using the distance formula.
Step-by-step explanation:
To find the constant a so that the point P=(5, a) is equidistant from points A=(1,1) and B=(7,-2), we need to use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula d = sqrt((x2 - x1)^2 + (y2 - y1)^2). To make the distance from P to A equal to the distance from P to B, we can set up the following equation: sqrt((5 - 1)^2 + (a - 1)^2) = sqrt((5 - 7)^2 + (a - (-2))^2).
Simplifying the equation gives: sqrt(16 + (a - 1)^2) = sqrt(4 + (a + 2)^2). Squaring both sides of the equation eliminates the square root: 16 + (a - 1)^2 = 4 + (a + 2)^2.
Expanding and simplifying the equation leads to: a^2 - a - 27 = 0. Solving this quadratic equation gives two possible values for a: a = -3 or a = 9.