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Positional relationship between a parabola and a circle

Question:

The equation of a parabola is, y2 = - 8x. The focus of the parabola is the center of a circle. The circle passes the origin. What is the equation of the circle?

Solution:

The parabola has the equation, y2 = - 8x, when x = 0, y = 0, so, the parabola passes the origin. Because there is a negative sign, so, the range of x must be negative and including the zero point. so, the graph of the parabola opens left. So, the standard parabola equation is, y2 = - 2px (p > 0). Now, we determine the value of p.

y2 = - 8x (given)
y2 = - 2px (parabola equation)
2p = 8
p = 4

The focus of the parabola is F (-p/2, 0). So, the focus of this parabola is F (-2, 0). The x-coordinate of the focus of the parabola is -2 and y-coordinate of the focus of the parabola is 0. Because the focus of the parabola is the center of the circle. So, we get the coordinate of the center of the circle which is (-2, 0). Because the circle passes the origin, so, the radius of the circle is 2. Let r be the radius of the circle, so, r = 2. So, the equation of the circle is,

[x - (-2)]2 + (y - 0)2 = 22
(x + 2)2 + y2 = 4

Therefore, the equation of the circle is, (x + 2)2 + y2= 4. Watch the video for more details.

how to find the circle equation from a parabola?

Look the figure above, the blue curve is the parabola, y2 = -8x. Its vertex is (0, 0) and it open left. The focus of the parabola is (-2, 0). The red curve is the circle, its center is (-2, 0) and the radius of the circle is r = 2. The equation of the circle is (x + 2)2 + y2 = 4.