Respuesta :
Answer:
Step-by-step explanation:
Suppose we a point [tex]P(x,y,z)[/tex] such that its distance from either the point [tex]A(3,4,-5)[/tex] or [tex]B(-2,1,4)[/tex] is the same.
Using this information we can formula:
distance AP = distance BP
first, let's find the distance from AP, using the distance formula.
[tex]r = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2}[/tex]
[tex]AP = \sqrt{(3 - x_2)^2 + (4 - y_2)^2 + (-5 - z_2)^2}[/tex]
similarly, we can find the distance BP
[tex]BP = \sqrt{(-2 - x_2)^2 + (1 - y_2)^2 + (4 - z_2)^2}[/tex]
since both distances are exactly the same we can equate them
[tex]AP = BP[/tex]
[tex]\sqrt{(3 - x_2)^2 + (4 - y_2)^2 + (-5 - z_2)^2} = \sqrt{(-2 - x_2)^2 + (1 - y_2)^2 + (4 - z_2)^2}[/tex]
we can simplify it a bit squaring both sides, and getting rid of the subscripts.
[tex](3 - x)^2 + (4 - y)^2 + (-5 - z)^2 = (-2 - x)^2 + (1 - y)^2 + (4 - z)^2[/tex]
what we have done here is formulated an equation which consists of any point P that will have the same distance from (3,4,-5) and (-2,1,4).
To put it more concretely,
This is the equation of the the plane from that consists of all points (P) from which the distance from both (3,4,-5) and (-2,1,4) are equal.
