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 standard hyperbola equation when its focus lies on y-axis

What is the standard hyperbola equation when its focus located in y-axis?

Equation for the asymptotes of the standard hyperbola when its focus lies on y-axis

How to find the equation for asymptotes of the hyperbola when its focus lies on the y-axis?

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

what are the graph of the hyperbola and the graph of 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).