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