Question 1121260
.
Find the Standard equation of hyperbola, center, foci, vertices at asymptotes of the function 4x² - 5y² + 32x + 30y = 1.
Thank you!
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<pre>
4x^2 - 5y^2 + 32x + 30y = 1  ====>  Complete the squares for x- and y-terms separately  ====>  


(4x^2 + 32x) + (-5y^2 + 30y) = 1


4(x^2 + 8x) - 5*(y^2 - 6y) = 1


4*(x^2 + 8x + 16) - 5*(y^2 - 6y + 9) = 1 + 4*16 - 5*9


4*(x+4)^2 - 5*(y-3)^2 = 20.     <<<---=== Divide by 20 both sides


{{{(x+4)^2/5}}} - {{{(y-3)^2/4}}} = 1


{{{(x+4)^2/(sqrt(5))^2}}} - {{{(y-3)^2/2^2}}} = 1


It is the standard form equation for a hyperbola.


The hyperbola has the center at  (x,y) = (-4,3).


Real axis is parallel to x-axis; imaginary axis is parallel to y-axis.


Real semi-axis is {{{sqrt(5)}}} units long;  Imaginary axis is 2 units long.


Vertices are at y= 3:  x = {{{-4 - sqrt(5)}}}  and  x= {{{-4 + sqrt(5)}}}.


The foci are  x= {{{-4 - sqrt(5+2^2)}}} = -4 - 3 = -7  and  x= {{{-4 + sqrt(5+2^2)}}} = -4 + 3 = -1.
</pre>

See the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-definition--canonical-equation--characteristic-points-and-elements.lesson>Hyperbola definition, canonical equation, characteristic points and elements</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-focal-property.lesson>Hyperbola focal property</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-and-normal-vectors-to-a-hyperbola.lesson>Tangent lines and normal vectors to a hyperbola</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-a-hyperbola.lesson>Optical property of a hyperbola</A> 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-a-hyperbola.lesson>Standard equation of a hyperbola</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-hyperbola-given-by-its-standard-eqn-NEW.lesson>Identify elements of hyperbola given by its standard equation</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-the-standard-equation-of-a-hyperbola-given-by-its-elements.lesson>Find the standard equation of a hyperbola given by its elements</A> 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-a-hyperbola.lesson>General equation of a hyperbola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-a-hyperbola-to-the-standard-form-by-completing-the-square.lesson>Transform general equation of a hyperbola to the standard form by completing the square</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-hyperbola-given-by-its-gen-eqn.lesson>Identify elements of a hyperbola given by its general equation</A> 

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic 
"<U>Conic sections: Hyperbolas. Definition, major elements and properties. Solved problems</U>".



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.