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# Application of parabola and hyperbola

Question

If the focus of parabola x2 = 4py (p > 0) is overlap with one of the focus of hyperbola y2/5 - x2/4 = 1. What is the value of p?

Solution

### The graph of the asymptotes of the hyperbola

In the figure above, the center of the rectangle is the origin. The width of the rectangle is 2b and the height of the rectangle is 2a. The diagonals of the rectangle are the asymptotes of the hyperbola. One of the asymptote of the hyperbola is y = (a/b) x and another is y = -(a/b) x. The graph of the hyperbola is never across its asymptotes.

### The intersection points of the hyperbola with axis

The points that the hyperbola intersect with y-axis

Let x = 0, then y2/5 = 1, then y2 = 5. Then y1 = -square root of 5 = -2.24 and y2 = square root of 5 = 2.24. Then one of the vertex of the hyperbola is (0, -2.24) and another is (0, 2.24).

Is the hyperbola intersect with x-axis?

Let y = 0, then -x2/4 = 1, then x2 = -4, x = +- 2i. These is no real solution, so the graph of the hyperbola is never intersect with x-axis. Therefore, x-axis is an imaginary axis and y-axis is a real axis.

### What are the focus of the given hyperbola?

From the given equation of the hyperbola, y2/5 - x2/4 = 1. We obtained a2 = 5 and b2 = 4. For hyperbola, c2 = a2 + b2 = 5 + 4 = 9. So c1 = -3 and c2 = 3. Therefore, one coordinate of the focus of the hyperbola is F1(0, -3) and another is F2(0, 3)

Given the focus of the parabola x2 = 4py (p > 0) overlap with one of the focus of the hyperbola.

The graph of the parabola x2 = 4py (p > 0) is symmetry to the y-axis. Its vertex is (0, 0) and the graph open up. Therefore the focus of the parabola should be F (0, 3).

When the equation of the parabola is x2 = 4py (p > 0), its focus is F (0, p). Therefore, p = 3.

The equation of the parabola is x2 = 4py = 4(3) y = 12y

### The graph of the hyperbola and the parabola

In the figure above, the blue curve is one branch of the hyperbola and the orange curve is another branch of the hyperbola. The hyperbola (orange curve) intersect y-axis is y1 = -2.24 and the hyperbola (blue curve) intersects with y-axis is y2 = 2.24. The gray curve is the parabola which open up and has the vertex (0, 0).