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# Example of a circle and a parabola

Question:

Point M (2, n) n > 0, lies on a parabola y2 = 2px (p > 0). The distance from point M to the focus of the parabola is 4. If M is the center of the circle and the circle is tangent to the x-axis, then what is the equation of the circle?

Solution:

For the parabola y2 = 2px (p > 0), when x = 0, y = 0. The parabola passes the origin. For each x value, there are a positive and a negative y value, so, the parabola is symmetry to the x-axis. The larger the value of x, the larger the absolute value of y, so, the graph is open right.

The focus of the parabola y2 = 2px (p > 0) is F (p/2, 0).

The directrix of the parabola y2 = 2px (p > 0) is x = -p/2. M is a point lies on the parabola. M has the coordinate (2, n) n > 0. Now draw a line passes the point M and perpendicular to the directrix of the parabola, this line intersects the directrix at the point G. Connect MG and connect MF. The distance from the point M to the point G is d. d = x coordinate of the point M subtract the x-coordinate of the point G. d = 2 - (-p/2) = 2 + p/2. We are given the distance from the point M to the focus of the parabola is 4. That is, |MF| = 4. Using the property of a parabola, the distance from any point on a parabola to the focus is equal to the distance from that point to the directrix of the parabola.

|MF| = |MG|
MF = 4 (given), MG = d = 2 + p/2
2 + p/2 = 4
p/2 = 2
p = 4

The parabola equation is y2 = 2py = 2 (4) x = 8x

Because the point M (2, n) lies on the parabola, so the point M (2, n) satisfy the parabola equation. Now, substitute the point M into the parabola equation.

y2 = 8x
n2 = 8 (2) = 16
n = +- 4
given n > 0
n = 4

The coordinate of the point M is (2, 4). Given the point M is the center of the circle. Because the circle is tangent to the x-axis, so, the y-coordinate of the point M is the radius of the circle. That is, r = 4. Therefore, the circle equation is: (x - 2)2 + (y - 4)2 = 16. Watch the video for more details.