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How to find the equation of the parabola?

Question:

The directrix of a parabola, x2 = 2py (p > 0), is tangent to a circle, x2 + y2 - 4y - 5 = 0, what is the equation of the parabola?

Solution:

The parabola has the equation, x2 = 2py (p > 0). When x = 0, y = 0, so the parabola passes the origin. When x is negative, x square is positive, so, y is positive. When x is positive, y is positive. The larger the absolute value of x is, the large of the y is. So the graph of the parabola is open up and symmetry to the positive y-axis.

For the parabola, x2 = 2py (p > 0), its directrix equation is, y = - p/2. We need to find the value of p.

We need to find the intersection point of the directrix and y-axis. We are given that the circle is tangent to the directrix of the parabola. So, we need to find the center of the circle and the radius of the circle.

x2 + y2 - 4y - 5 = 0
x2 + y2 - 4y + 22 - 22 - 5 = 0
x2 + (y - 2)2 = 22 + 5
x2 + (y - 2)2 = 4 + 5
x2 + (y - 2)2 = 9
x2 + (y - 2)2 = 32

The coordinate of the center of the circle is (0, 2) and the radius of the circle is 3

Now we find that the circle intersects y-axis at y = -1, this line is the directrix of the parabola. So, -P/2 = -1. We get P = 2. Therefore, the equation of the parabola is: x2 = 4y. Watch the video for more details.