Determine the shortest distance from point A (-5, -2sqrt7) to the X-axis to point B ( 4, -sqrt7).
Hello, strwbrry869!
There is a back-door approach to this problem.Determine the shortest distance from point to the x-axis to pointCode:| _ | (4,√7) | o B' | * : -5 | P * : ---+--------------+--o--------+---- : * * : : * | * : : * | o B_ : * | (4,-√7) : * | A o _ | (-5,-2√7) |
Reflect point over the -axis to point
Draw line , intersecting the -axis at
gives the shortest distance from to plus from to
. . (Do you see why?)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Using calculus, the problem gets VERY messy!
The distance from to is:
.
The distance from to is:
. .
The total distance is: .
And that is the function we must minimize . . .
Continuing the Calculus way (the best I can):
The distance formula, , looks right. I think from there you would have to take the derivative of that and set it equal to 0 to find the minimum value.
So:
To find the minimum, we substitute 0 for .
That is an extremely difficult equation to factor, so I just put it in my calculator and used the Equation Solver.
I found that:
So, I guess that that (0.3411, 0), would be your point of intersection on the x-axis.
So you can now just find the distance between A and the point of intersection, and then the distance between B and the point of intersection, and add them together.
Hope that helps a little. Seems like an extremely hard problem for an Algebra student. I'm in Calculus now, and that seems pretty difficult to me.