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# Application of parabola and hyperbola

# The standard hyperbola equation when its focus lies on y-axis

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

# The graph of the asymptotes of the hyperbola

# The intersection points of the hyperbola with axis

# What are the focus of the given hyperbola?

# The graph of the hyperbola and the parabola

**Question**

If the focus of parabola x^{2} = 4py (p > 0) is overlap with one of the focus of hyperbola y^{2}/5 - x^{2}/4 = 1. What is the value of p?

**Solution**

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 points that the hyperbola intersect with y-axis**

Let x = 0, then y^{2}/5 = 1, then y^{2} = 5. Then y_{1} = -square root of 5 = -2.24 and y_{2} = 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 -x^{2}/4 = 1, then x^{2} = -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.

From the given equation of the hyperbola, y^{2}/5 - x^{2}/4 = 1. We obtained a^{2} = 5 and b^{2} = 4. For hyperbola, c^{2} = a^{2} + b^{2} = 5 + 4 = 9.
So c_{1} = -3 and c_{2} = 3. Therefore, one coordinate of the focus of the hyperbola is F_{1}(0, -3) and another is F_{2}(0, 3)

**Given the focus of the parabola x ^{2} = 4py (p > 0) overlap with one of the focus of the hyperbola.**

The graph of the parabola x^{2} = 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 x^{2} = 4py (p > 0), its focus is F (0, p). Therefore, p = 3.

The equation of the parabola is x^{2} = 4py = 4(3) y = 12y

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 y_{1} = -2.24 and the hyperbola
(blue curve) intersects with y-axis is y_{2} = 2.24. The gray curve is the parabola which open up and has the vertex (0, 0).