5x(1+1/x^2+y^2)=12 ,5y(1-1/x^2+y^2)=4 solve it for x,y
Answers
Given : 5x(1+1/(x²+y²))=12 , 5y(1-1/(x²+y²))=4
To find : solve it for x,y
Solution:
5x(1+1/(x²+y²))=12
5y(1-1/(x²+y²))=4
Let say x²+y² = a
=> 5x ( 1 + 1/a) = 12 => 5x = 12a/(a + 1)
& 5y ( 1- 1/a) = 4 => 5y = 4a/(a - 1)
Squaring and adding
=> (5x)² + (5y)² = (12a/(a + 1))² + (4a(a - 1))²
=> 25a ( a² - 1)² = 144a² (a - 1)² + 16a²(a + 1)²
=> 25 (a⁴ - 2a² + 1) = 144a (a² - 2a + 1) + 16a(a² + 2a + 1)
=> 25a⁴ - 50a² + 25 = 160a³ - 256a² + 160a
=> 25a⁴ - 160a³ + 206a² - 160a + 25 = 0
=> (a - 5)(5a - 1) (5a² - 6a + 5) = 0
a = 5 or a= 1/5 or 5a² - 6a + 5 leads to complex number 0.6 ± 0.8i
5x = 12a/(a + 1) => 5x = 10 => x = 2 or 5x = 2 => x = 2/5
5y = 4a/(a - 1) => 5y = 5 => y = `1 or 5y = -1 => y = -1/5
x , y = ( 2, 1) or ( 2/5 , -1/5)
Learn more:
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answer:
5x(1+1/(x²+y²))=12
5y(1-1/(x²+y²))=4
Let say x²+y² = a
=> 5x ( 1 + 1/a) = 12 => 5x = 12a/(a + 1)
& 5y ( 1- 1/a) = 4 => 5y = 4a/(a - 1)
Squaring and adding
=> (5x)² + (5y)² = (12a/(a + 1))² + (4a(a - 1))²
=> 25a ( a² - 1)² = 144a² (a - 1)² + 16a²(a + 1)²
=> 25 (a⁴ - 2a² + 1) = 144a (a² - 2a + 1) + 16a(a² + 2a + 1)
=> 25a⁴ - 50a² + 25 = 160a³ - 256a² + 160a
=> 25a⁴ - 160a³ + 206a² - 160a + 25 = 0
=> (a - 5)(5a - 1) (5a² - 6a + 5) = 0
a = 5 or a= 1/5 or 5a² - 6a + 5 leads to complex number 0.6 ± 0.8i
5x = 12a/(a + 1) => 5x = 10 => x = 2 or 5x = 2 => x = 2/5
5y = 4a/(a - 1) => 5y = 5 => y = `1 or 5y = -1 => y = -1/5
x , y = ( 2, 1) or ( 2/5 , -1/5)