Answer:
P(1,2)
Step-by-step explanation:
There are 2 points.
A(-2,4) and B(7,6)
the point P on the y=2 can also represented as P(x,2)
We can use the distance formula to find the distances AP and BP
[tex]\text{dist} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
for AP: A(-2,4) and P(x,2)
[tex]AP = \sqrt{(-2 - x)^2 + (4 - 2)^2}[/tex]
[tex]AP = \sqrt{(-2 - x)^2 + 4}[/tex]
[tex]AP = \sqrt{(-1)^2(2 + x)^2 + 4}[/tex]
[tex]AP = \sqrt{(2 + x)^2 + 4}[/tex]
for BP: B(7,6) and P(x,2)
[tex]BP = \sqrt{(7 - x)^2 + (6 - 2)^2}[/tex]
[tex]BP = \sqrt{(7 - x)^2 + 16}[/tex]
the total distance AP + BP will be
[tex]\sqrt{(2 + x)^2 + 4}+\sqrt{(7 - x)^2 + 16}[/tex] (plot is given below)
Our task is to find the value of x such that the above expression is small as possible. (we can find this either through plotting or differentiating)
If you plot the above equation, the minimum point of the curve will be clearly visible, and it will be at x = 1. Hence, the point P(1,2) is such that the total distance AP + BP is as small as possible.
