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Hi can I please have help with this problem? Thank you!!

Hi can I please have help with this problem? Thank you!!-example-1
User Zett
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Answer:

Hyperbola (vertical)

Explanation:

Given conic section equation:


5x^2-11y^2-26x+2y+36=0

The general equation for any conic section is:


\boxed{Ax^2+Bxy+Cy^2+Dx+Ey+F=0}

where A, B, C, D, E and F are constants.

If B² − 4AC < 0, if a conic exists, it will be either a circle or an ellipse.
(If A = C and B = 0, the equation represents a circle).

If B² − 4AC = 0, if a conic exists, it will be a parabola.

If B² − 4AC > 0, if a conic exists, it will be a hyperbola.

For the given equation, the coefficients of A, B and C are:

  • A = 5
  • B = 0
  • C = -11

Substitute the coefficients into B² − 4AC:


\begin{aligned}B^2-4AC&amp;=0^2-4(5)(-11)\\&amp;=0-20(-11)\\&amp;=220 > 0\end{aligned}

As B² − 4AC > 0, the conic is a hyperbola.


\hrulefill

To rewrite the given equation in the standard form of a hyperbola, we can complete the square.

Given equation:


5x^2-11y^2-26x+2y+36=0

Arrange the equation so all the terms with variables are on the left side and the constant is on the right side:


5x^2-26x-11y^2+2y=-36

Factor out the coefficient of the x² term and the coefficient of the y² term:


5\left(x^2-(26)/(5)x\right)-11\left(y^2-(2)/(11)y\right)=-36

Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:


5\left(x^2-(26)/(5)x+(169)/(25)\right)-11\left(y^2-(2)/(11)y+(1)/(121)\right)=-36+(169)/(5)-(1)/(11)

Factor the two perfect trinomials on the left side and simplify the right side:


5\left(x-(13)/(5)\right)^2-11\left(y-(1)/(11)\right)^2=-(126)/(55)

Divide both sides by the number of the right side so the right side equals 1:


(5\left(x-(13)/(5)\right)^2)/(-(126)/(55))-(11\left(y-(1)/(11)\right)^2)/(-(126)/(55))=(-(126)/(55))/(-(126)/(55))


(-\left(x-(13)/(5)\right)^2)/((126)/(275))+(\left(y-(1)/(11)\right)^2)/((126)/(605))=1


(\left(y-(1)/(11)\right)^2)/((126)/(605))-(\left(x-(13)/(5)\right)^2)/((126)/(275))=1

As the y²-term is positive, the hyperbola is vertical (opening up and down).

User Mockobject
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