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.