write an equation which must be true if the line from p(x,y) to a(25,0) is perpendicular to the line from P(x,y) to the center of the circle x^2+y^2=400
To find the equation that must be true for the line from point P(x,y) to A(25,0) to be perpendicular to the line from P(x,y) to the center of the circle x^2 + y^2 = 400, we need to utilize the concept of slope and the perpendicularity condition.
First, let's find the slope of the line from P(x,y) to the center of the circle. The center of the circle is the origin (0,0), so the slope of this line is simply (change in y) / (change in x) = y / x.
Next, let's find the slope of the line from P(x,y) to A(25,0). The slope of this line is (change in y) / (change in x) = (0 - y) / (25 - x) = -y / (25 - x).
For two lines to be perpendicular, their slopes must be negative reciprocals of each other. Therefore, we can set up the following equation:
y / x = -(25 - x) / y
To simplify this equation and remove the fractions, we can cross multiply:
y^2 = -x(25 - x)
Now, let's simplify further:
y^2 = -25x + x^2
Finally, we can rearrange the equation to get it in standard form:
x^2 + 25x + y^2 = 0
Therefore, the equation that must be true for the line from P(x,y) to A(25,0) to be perpendicular to the line from P(x,y) to the center of the circle x^2 + y^2 = 400 is x^2 + 25x + y^2 = 0.