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 $\displaystyle A(\text{-}5,\:\text{-}2\sqrt{7})$ to the x-axis to point $\displaystyle B(4,\:\text{-}\sqrt7)$Code:| _ | (4,√7) | o B' | * : -5 | P * : ---+--------------+--o--------+---- : * * : : * | * : : * | o B_ : * | (4,-√7) : * | A o _ | (-5,-2√7) |
Reflect point $\displaystyle B$ over the $\displaystyle x$-axis to point $\displaystyle B'.$
Draw line $\displaystyle AB'$, intersecting the $\displaystyle x$-axis at $\displaystyle P.$
$\displaystyle P$ gives the shortest distance from $\displaystyle A$ to $\displaystyle P$ plus from $\displaystyle P$ to $\displaystyle B.$
. . (Do you see why?)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Using calculus, the problem gets VERY messy!
The distance from $\displaystyle A(-5,-2\sqrt{7})$ to $\displaystyle P(x,y)$ is:
.$\displaystyle d_1 \:=\:\sqrt{(x+5)^2 + (2\sqrt{7})^2} \:=\:\sqrt{x^2 + 10x + 53}$
The distance from $\displaystyle B(4,-\sqrt{7})$ to $\displaystyle P(x,y)$ is:
. . $\displaystyle d_2 \;=\;\sqrt{(x-4)^2 + (\sqrt{7})^2} \:=\:\sqrt{x^2+10x + 23} $
The total distance is: .$\displaystyle D \;=\;\sqrt{x^2+10x + 53} + \sqrt{x^2 - 8x + 23}$
And that is the function we must minimize . . .
Continuing the Calculus way (the best I can):
The distance formula, $\displaystyle D \;=\;\sqrt{x^2+10x + 53} + \sqrt{x^2 - 8x + 23}$, 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:
$\displaystyle
\frac{dD}{dx} = \frac{1}{2}(x^2+10x+53)^\frac{-1}{2}*(2x+10) + \frac{1}{2}(x^2+10x+23)^\frac{-1}{2}*(2x-8)
$
$\displaystyle
\frac{dD}{dx} = \frac{2x+10}{2\sqrt{x^2+10x+53}} + \frac{2x-8}{2\sqrt{x^2+10x+23}}
$
To find the minimum, we substitute 0 for $\displaystyle \frac{dD}{dx}$.
$\displaystyle
0 = \frac{2x+10}{2\sqrt{x^2+10x+53}} + \frac{2x-8}{2\sqrt{x^2+10x+23}}
$
That is an extremely difficult equation to factor, so I just put it in my calculator and used the Equation Solver.
I found that:
$\displaystyle
x = 0.34110898362998....
$
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.