1. Find the exact solutions of x^2 - (y-12)^2 = 144 and y = -x^2
Is the answer no solution?
2. Solve the system: y^2 - 4x^2 = 4 and y = 2x
Solving this myself I got no solution, but someone who helped me with this question got the answer x= + i sqrt 2, -i sqrt 2
1. They do not intersect
http://www.wolframalpha.com/input/?i=plot+x%5E2+-+(y-12)%5E2+%3D+144+and+y+%3D+-x%5E2
2. Just like in #1, there is no REAL solution but both have an imaginary or complex solution
y^2 - 4x^2 = 4 and y = 2x , use substitution
(2x)^2 - 4x^2 = 4
0 = 4 , contradiction, no solution at all
interesting situation,
y^2/4 - x^2 = 1
or
x^2 - y^2/4 = -1 is a vertical hyperbola
the asymptotes are x-y/2 = 0 and x + y/2 = 0
x - y/2 = 0 ----> y = 2x <------- the given line, hyperbolas reach their asymptotes at infinity
1. x^2 - (y-12)^2 = 144, and Y = -x^2.
x^2 - (y^2 - 24y + 144) = 144,
Replace x^2 with -y:
-y - y^2 + 24y -144 = 144,
-y^2 + 23y - 144 = 144,
-y^2 + 23y - 288 = 0,
Use Quadratic Formula:
Y = (-23 +- sqrt(529 - 1152))/(-2) ,
Y = (-23 +- 24.96i)/(-2) = 11.5 - 12.5i, and 11.5 + 12.5i.
So we have 2 imaginary solutions.
To find the solutions to the given equations, we'll go step by step.
1. Find the exact solutions of x^2 - (y-12)^2 = 144 and y = -x^2:
First, substitute y = -x^2 into the first equation:
x^2 - (-x^2 - 12)^2 = 144
x^2 - (x^2 + 12)^2 = 144
Expand the equation:
x^2 - (x^4 + 24x^2 + 144) = 144
x^2 - x^4 - 24x^2 - 144 = 144
-x^4 - 25x^2 - 288 = 0
This equation is a polynomial of degree 4. To solve it, you can use numerical methods or graphing calculators/software. Unfortunately, the equation cannot be easily solved by factoring or using algebraic methods. So we cannot find exact solutions to this equation analytically.
Therefore, the answer to the question is "no solution" since we are unable to find any exact solution to this equation.
2. Solve the system: y^2 - 4x^2 = 4 and y = 2x:
Substitute y = 2x into the first equation:
(2x)^2 - 4x^2 = 4
4x^2 - 4x^2 = 4
0 = 4
This equation is contradictory. It indicates that there is no value of x that will satisfy both equations simultaneously.
So, you were correct in stating that there is no solution to this system of equations.
It's important to note that the person who helped you with this question may have made an error in their calculation, as the equation does not yield x= + i sqrt 2 or -i sqrt 2 as solutions.