(sqrt(x^2+y^2+100x+2500))-(sqrt(x^2+y^2-100x+2500))=60
√(x^2+y^2+100x+2500)-√(x^2+y^2-100x+2500) = 60
(x^2+y^2+100x+2500)-2√((x^2+y^2+100x+2500)(x^2+y^2-100x+2500))+(x^2+y^2-100x+2500) = 3600
2(x^2+y^2+2500) - 2√(x^4+2x^2y^2+y^4-5000x^2+5000y^2+6250000) = 3600
√(x^4+2x^2y^2+y^4-5000x^2+5000y^2+6250000) = x^2+y^2+700
x^4+2x^2y^2+y^4-5000x^2+5000y^2+6250000 = x^4+2x^2y^2+y^4+1400x^2+1400y^2+490000
-5000x^2+5000y^2+6250000 = 1400x^2+1400y^2+490000
-6400x^2+3600y^2+5760000 = 0
64x^2-36y^2 = 57600
x^2/900 - y^2/1600 = 1
That's just an hyperbola with vertices at (±30,0)
To solve the equation (sqrt(x^2+y^2+100x+2500))-(sqrt(x^2+y^2-100x+2500))=60, we need to isolate one of the square roots and then square both sides to eliminate the square roots.
Step 1: Isolate one square root term
(sqrt(x^2+y^2+100x+2500))-(sqrt(x^2+y^2-100x+2500))=60
Let's isolate the second square root term by moving it to the right side of the equation:
(sqrt(x^2+y^2+100x+2500)) = 60 + (sqrt(x^2+y^2-100x+2500))
Step 2: Square both sides of the equation
To eliminate the square roots, we need to square both sides of the equation:
[(sqrt(x^2+y^2+100x+2500))]^2 = [(60 + sqrt(x^2+y^2-100x+2500))]^2
Simplifying the left side:
(x^2+y^2+100x+2500) = [(60 + sqrt(x^2+y^2-100x+2500))]^2
Step 3: Expand the right side of the equation
[(60 + sqrt(x^2+y^2-100x+2500))]^2 can be expanded as:
[(60 + sqrt(x^2+y^2-100x+2500))][(60 + sqrt(x^2+y^2-100x+2500))]
Using the FOIL method (First, Outer, Inner, Last):
= 60 * 60 + 60 * sqrt(x^2+y^2-100x+2500) + 60 * sqrt(x^2+y^2-100x+2500) + (sqrt(x^2+y^2-100x+2500)) * (sqrt(x^2+y^2-100x+2500))
Simplifying further:
= 3600 + 120 * sqrt(x^2+y^2-100x+2500) + (x^2+y^2-100x+2500)
= 3600 + (x^2 - 100x + y^2) + 120 * sqrt(x^2+y^2-100x+2500)
Simplifying the right side:
(x^2+y^2+100x+2500) = 3600 + (x^2 - 100x + y^2) + 120 * sqrt(x^2+y^2-100x+2500)
Step 4: Simplify the equation
Now we can simplify the equation by canceling out the like terms on both sides:
(x^2+y^2+100x+2500) = (x^2 - 100x + y^2) + 120 * sqrt(x^2+y^2-100x+2500)
The x^2 and y^2 terms on both sides cancel out:
100x + 2500 = -100x + 120 * sqrt(x^2+y^2-100x+2500)
Step 5: Solve for x
Now we have a simple linear equation that can be solved for x:
200x = 120 * sqrt(x^2+y^2-100x+2500) - 2500
200x + 2500 = 120 * sqrt(x^2+y^2-100x+2500)
To isolate the square root term on the right side, we will square both sides of the equation:
(200x + 2500)^2 = [120 * sqrt(x^2+y^2-100x+2500)]^2
Expanding both sides of the equation:
40000x^2 + 1000000 + 100000x = 14400(x^2+y^2-100x+2500)
40000x^2 + 1000000 + 100000x = 14400x^2 + 14400y^2 - 1440000x + 36000000
Rearranging the terms:
0 = 10400x^2 + 14400y^2 - 1544000x + 35000000
Step 6: Simplify the equation
Finally, we have a quadratic equation in terms of x and y:
10400x^2 + 14400y^2 - 1544000x + 35000000 = 0
This equation can be solved using various methods such as factoring, completing the square, or using the quadratic formula.