Classify each conic section and write its equations in standard form. show work.

The conic is parabola, its equation is (y + 5)² = -(x + 1)

Step-by-step explanation:

- The general equation for any conic section is  

Ax² + Bxy + Cy² + Dx + Ey + F = 0  

where A , B , C , D , E and F are constants. A, B, and C are not all zero

- When we change the values of some of the constants, the shape  

of the corresponding conic will also change.  

- It is important to know the differences in the equations to  

identify the type of conic that is represented by a given equation.  

# If B² − 4AC is less than zero, if a conic exists, it will be either a  

circle or an ellipse  

# If B² − 4AC equals zero, if a conic exists, it will be a parabola  

# If B² − 4AC is greater than zero, if a conic exists, it will be a  

hyperbola  

* How to identify the type of the conic

- Rewrite the equation in the general form,  

Ax² + Bxy + Cy² + Dx + Ey + F = 0  

- Identify the values of A and C from the general form.  

- If A and C are nonzero, have the same sign, and are not equal

to each other, then the graph is an ellipse.

- If A and C are equal and nonzero and have the same sign, then

the graph is a circle  

- If A and C are nonzero and have opposite signs, and are not equal

then the graph is a hyperbola.

- If either A or C is zero, then the graph is a parabola

* Now lets solve the problem

- The equation is y² + x + 10y + 26 = 0

 A = 0 , B = 0 , C = 1 , D = 1 , E = 10 and F = 26

∵ A = 0

∴ The equation is a parabola its standard form is:

  (y - k)² = 4 p (x - h), where (h , k) is the vertex point of the parabola,

   with a horizontal axis y = k

- Lets change the standard form to the general form

∵ (y - k)² = 4 p (x - h) ⇒ open the brackets

∴ y² - 2ky + k² = 4px - 4ph

- Put all of them in one side

∴ y² - 4px - 2ky + k² + 4ph = 0

- Compare it with the equation

  y² - 4px - 2ky + k² + 4ph = 0 ⇒ y² + x + 10y + 26 = 0

∵ -4px = 1x ⇒ cancel x

∴ -4p = 1 ⇒ divide both sides by -4

∴ p = -1/4

∵ -2ky = 10y ⇒ cancel y

∴ -2k = 10 ⇒ divide both sides by -2

∴ k = -5

∵ k² + 4ph = 26 ⇒ (-5)² + 4(-1/4)h = 26 ⇒ simplify

∴ 25 - h = 26 ⇒ subtract 25 from both sides

∴ h = -1

- Now we can write the standard form of the equation

∴ (y - -5)² = 4(-1/4)(x - -1)

∴ (y + 5)² = -(x + 1)

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step-by-step explanation: