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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.

circle and parabola and find circle equation

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.

find circle equation when center of the circle lies on a parabola.

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.

find circle equation when center of the circle lies on a parabola.

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