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# Find the equation of a parabola

Question:

The vertex of a parabola is in the origin. The parabola is symmetry to the y-axis. The focus of the parabola lies in the line 2x - 3y + 6 = 0. Find the equation of the parabola.

Solution:

Because the vertex of a parabola is in the origin. The parabola is symmetry to the y-axis. So, the parabola may have the equation x2 = 2py (p > 0) and the graph of the parabola is open upward. Or the parabola may have the equation x2 = -2py (p > 0) and the graph of the parabola is open downward. We need another condition to determine which parabola equation to use.

Given that the focus of the parabola lies in the line 2x - 3y + 6 = 0.

Now, draw the graph of the line 2x - 3y + 6 = 0. When x = 0, y = 2. When y = 0, x = -3. So, the line passes these two points (-3, 0) and (0, 2).

Because the parabola is symmetry to the y-axis, so, the focus of the parabola lies on the y-axis. The focus of the parabola is F (0, 2).

So, the parabola equation is x2 = 2py (p > 0) and the graph is open upward. In this case, the coordinate of the focus is F (0, p/2).

So, we get the equation p/2 = 2 -> p = 4. So, the parabola equation is x2 = 2py = 2 × 4y = 8y.

In the figure above, the blue curve is the parabola x2 = 8x and the line has the equation 2x - 3y + 6 = 0. The parabola has the focus F (0, 2). Watch the video for more details.