SOLUTION: Sketch and identify the given conic 12 x^2 – 3y^2 – 18x - 3y – 10 = 0

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Sketch and identify the given conic 12 x^2 – 3y^2 – 18x - 3y – 10 = 0      Log On


   

Question 129870: Sketch and identify the given conic


12 x^2 – 3y^2 – 18x - 3y – 10 = 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
12+x%5E2+%96+3y%5E2+%96+18x+-+3y+%96+10+=+0Start with the given equation


12x%5E2-18x-3y%5E2-3y-10=0 Rearrange the terms


12x%5E2-18x-3y%5E2-3y=%2B10 Add 10 to both sides


12%28x-3%2F4%29%5E2-27%2F4-3y%5E2-3y=%2B10 Complete the square for the x terms


12%28x-3%2F4%29%5E2-27%2F4-3%28y%2B1%2F2%29%5E2%2B3%2F4=%2B10 Complete the square for the y terms


12%28x-3%2F4%29%5E2-3%28y%2B1%2F2%29%5E2-24%2F4=%2B10 Combine like terms


12%28x-3%2F4%29%5E2-3%28y%2B1%2F2%29%5E2-6=%2B10 Reduce


12%28x-3%2F4%29%5E2-3%28y%2B1%2F2%29%5E2=%2B10%2B6 Add 6 to both sides


12%28x-3%2F4%29%5E2-3%28y%2B1%2F2%29%5E2=16 Combine like terms


-----------


Notice how the equation is now in the form %28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1. This means that this conic section is a hyperbola which opens up in the x direction.


So the conic section looks like this:



Graph of 12%28x-3%2F4%29%5E2-3%28y%2B1%2F2%29%5E2=16