SOLUTION: How would I write -4x^2 + 9y^2 + 32x + 36y - 64 = 0 in standard form? What are the steps?
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Question 1106320: How would I write -4x^2 + 9y^2 + 32x + 36y - 64 = 0 in standard form? What are the steps?
Found 3 solutions by Boreal, greenestamps, ikleyn:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
-4x^2 + 9y^2 + 32x + 36y - 64 = 0 ;
put positive square first and move constant to right side. Group x and y
9y^2+36y-4x^2+32x=64. Factor
9(y^2+4y)-4(x^2-8x)=64. Complete the square and add constants to right side
9(y^2+4y+4)-4(x^2-8x+16)=64+36-64
9(y+2)^2-4(x-4)^2=36. Divide both sides by 36
(y-(-2))^2/4-(x-4)^2/9=1, write in the form of (y-k)^2/b^2-(x-h)^2/a^2=1, equation of hyperbola.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
(1) complete the squares in both x and y; and
(2) divide by the appropriate constant to get "1" on the right hand side.
[original equation]
[group the x terms and y terms; move constant to other side]
[factor out the leading coefficients for both x and y]
[complete the squares in x and y, adding the same quantities to both sides of the equation]
[divide by the constant on the right to make the right side "1"]
[simplify and re-format]
[rearrange terms to get positive term first on left hand side]
[write in final standard form]
The equation is of a hyperbola with center (4,-2); asymptotes with slopes 2/3 and -2/3; branches opening upward and downward.
A graph: 2 branches of the hyperbola (red, green); asymptotes (blue, purple). The center is at the intersection of the asymptotes.
Answer by ikleyn(52802) (Show Source): You can put this solution on YOUR website!
.
It is a typical problem on conical sections.
See the lessons in this site, where you will find all instructions and explanations on how to solve such problems.
Learn it once for all.
- Hyperbola definition, canonical equation, characteristic points and elements
- Standard equation of a hyperbola
- Identify elements of hyperbola given by its standard equation
- Find the standard equation of a hyperbola given by its elements
- General equation of a hyperbola
- Transform general equation of a hyperbola to the standard form by completing the square
- Identify elements of a hyperbola given by its general equation
Also, you have this free of charge online textbook in ALGEBRA-II in this site
ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topics
"Conic sections: Hyperbolas. Definition, major elements and properties. Solved problems".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.
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