Find the Properties y^2-4y+6x-8=0

Problem

y2−4*y+6*x−8=0

Solution

1. Identify the type of conic section by observing that only one variable, y is squared, which indicates the equation represents a parabola.

2. Rearrange the equation to isolate the y terms on one side and the x and constant terms on the other.

y2−4*y=−6*x+8

3. Complete the square for the y terms by adding ((−4)/2)2=4 to both sides of the equation.

y2−4*y+4=−6*x+8+4

4. Factor the left side as a perfect square and simplify the right side.

(y−2)2=−6*x+12

5. Factor out the coefficient of x on the right side to put the equation in the standard form (y−k)2=4*p*(x−h)

(y−2)2=−6*(x−2)

6. Determine the vertex (h,k) from the standard form.

h=2

k=2

Vertex=(2,2)

7. Calculate the value of p by setting 4*p equal to the coefficient of the linear term.

4*p=−6

p=−1.5

8. Identify the orientation of the parabola. Since the y term is squared and p is negative, the parabola opens to the left.

9. Find the focus using the formula (h+p,k)

Focus=(2−1.5,2)=(0.5,2)

10. Find the directrix using the formula x=h−p

x=2−(−1.5)

x=3.5

11. Find the axis of symmetry using the formula y=k

y=2

Final Answer

Vertex: *(2,2), Focus: *(0.5,2), Directrix: *x=3.5, Axis of Symmetry: *y=2

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