What are the coordinates of the focus of the conic section shown below? y^2+16y-4x+4=0

Question

Find the focus.

Please show the process for finding p.

Answer 1

If all you want is the value of "p", simply solve for x in terms of y.

4x = y²+16y+4

x = (1/4)y²+4y+1

The coefficient of y² is equal to 1/4p.

So, 1/4p = 1/4

4p = 4

p=1

Since the coefficient of y² is positive, the parabola opens to the right. The directed distance from the vertex to the focus is positive, so p is +1.

On the way to finding p, you can complete the square at this step:

4x - 4 = y² + 16y +64 - 64

4x - 4 = (y+8)² - 64

4x = (y+8)² - 60

x = (1/4)(y+8)²-15

p still equals 1, but now you can identify the vertex as (-15, -8). That means the coordinates of the focus are (-14, -8), i.e. we added p to the x coordinate of the vertex.

desmos.com/calculator/aidle4ejsj

Use the slider on y₁ at the above graph to see that the distance from any point on the parabola is the same distance from the focus as it is from the directrix (the definition of a parabola).