Find the Properties x^2-2x-8y+17=0

Problem

x2−2*x−8*y+17=0

Solution

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

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

x2−2*x=8*y−17

3. Complete the square for the x terms by adding ((−2)/2)2=1 to both sides of the equation.

x2−2*x+1=8*y−17+1

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

(x−1)2=8*y−16

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

(x−1)2=8*(y−2)

6. Determine the vertex (h,k) by comparing the equation to the standard form.

h=1

k=2

Vertex=(1,2)

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

4*p=8

p=2

8. Identify the direction of opening based on the squared term and the sign of p Since x is squared and p>0 the parabola opens upward.

9. Find the focus using the formula (h,k+p) for a vertical parabola.

Focus=(1,2+2)

Focus=(1,4)

10. Find the directrix using the formula y=k−p

y=2−2

y=0

11. Find the axis of symmetry using the formula x=h

x=1

Final Answer

Vertex: *(1,2), Focus: *(1,4), Directrix: *y=0, Axis of Symmetry: *x=1

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