Algebra and square root simplification

[smart_worker]

Homework Statement

Points A and B are 10 km apart and it is determined from the sound of an explosion heard at those points at different times that the location of the explosion is 6 km closer to A than B. Show that the location of the explosion is restricted to a particular curve and find an equation of it.

Homework Equations

√((x − 5)² + y²) - √((x + 5)² + y²) = 6

The Attempt at a Solution

i must simplify this to get x²/9 - y²/16 = 1

it is a hyperbola i guess

i applied (a-b)^2 = a^2 -2ab +b^2

if i square both sides i am still getting square root because 2ab is twice the product of 1st and 2nd roots which is again a root

if i square both sides again i still get a root

if i bring one root to the other side and square both sides still i am getting a root

how do i cancel the roots

 

[haruspex]

√(x − 5)² + y² + √(x + 5)² + y² = 6

(For the purposes of the template, this is part of the problem statement, not a 'relevant equation'.)
I presume you mean √((x − 5)² + y²) + √((x + 5)² + y²) = 6
After squaring both sides there will only be one square root term. Isolate that on one side of the equation and square again.

 

[smart_worker]

Isolate that on one side of the equation and square again.

if i do that the result i am getting is
x⁴ + 51x² + 52y²x² + y² + y⁴ + 632 = 0

which is not a hyperbola i guess

 

[haruspex]

if i do that the result i am getting is
x⁴ + 51x² + 52y²x² + y² + y⁴ + 632 = 0

which is not a hyperbola i guess

Funny, I didn't notice the problem statement before - only saw the equation you posted. Did you add the problem statement in an edit? I may have had a stale window.
So actually the equation should be in the attempt at solution section.
Anyway, it is wrong. Think about the signs.

 

[LCKurtz]

if i do that the result i am getting is
x⁴ + 51x² + 52y²x² + y² + y⁴ + 632 = 0

which is not a hyperbola i guess

It isn't anything, since the left side is positive and the right side is zero.

 

[smart_worker]

Did you add the problem statement in an edit?

yes i did

i changed the signs,instead of + it must be -
but still the result i am getting is x^4 + 51x^2 + 52y^2 x^2 + y^2 + y^4 + 632 = 0

 

[haruspex]

yes i did

i changed the signs,instead of + it must be -
but still the result i am getting is x^4 + 51x^2 + 52y^2 x^2 + y^2 + y^4 + 632 = 0

Then you've made a mistake in the algebra. Starting from
√((x − 5)² + y²) - √((x + 5)² + y²) = 6
all the x⁴, y⁴ and x²y² terms should cancel.
If still stuck, please post all your working.

 

[Mark44]

The original equation in post #1 was edited, and is now as follows:

√((x − 5)² + y²) - √((x + 5)² + y²) = 6

This looks fine to me.

1. Moved one of the radicals to the other side and then square both sides.
2. Simplify the resulting equation. (Many terms drop out.)
3. Leave the radical where it is, and move the other terms to the other side.
4. Square both sides again.

The resulting equation is what you want.