Difference between Parabola and Hyperbola

Parabolas and hyperbolas are both types of conic sections, but they differ significantly in shape, properties, and real-world applications.

The fundamental difference is shown in the image below:

Parabola

A parabola is a U-shaped curve in which every point is equidistant from a fixed point called the focus and a fixed line called the directrix. Its characteristic U-shape can open in different directions depending on its equation.

Real-World Examples:

• Projectile motion: The path of a thrown ball follows a parabolic trajectory.
• Car headlights: Parabolic mirrors are used to direct light beams in a specific direction, ensuring focused illumination

Read More: Applications of Parabola in Real-Life

Hyperbola

A hyperbola consists of two separate curves, or branches, defined by the constant difference between the distance to two fixed points called foci. Unlike a parabola, a hyperbola has two focal points and is typically used to describe systems involving forces waves, and navigation.

Real-life Example :

• Navigation systems: GPS uses concepts based on hyperbolic geometry to determine positions by calculating distances from multiple satellites.
• Radio waves: Hyperbolic equations describe wave propagation patterns.

Read More: Real-Life Applications of Conic Sections

Parabola vs. Hyperbola: Key Differences

The following table summarizes the key differences between parabolas and hyperbolas:

Related Reads:

• Standard Equation of a Parabola
• Hyperbola Formula
• Hyperbolic Function

Solved Questions of Difference between Parabola and Hyperbola

Question 1: Find the coordinates of the focus and the equation of the directrix of the parabola: y²=16x.

Solution:

The standard form of a rightward-opening parabola is y²= 4ax

Comparing, 4a = 16⇒ a = 4.
Focus: (a, 0) = (4, 0)
Directrix: x = -a = -4

Question 2: Find the coordinates of the foci and the equations of the asymptotes for the hyperbola: \frac{x^2}{16} - \frac{y^2}{9} = 1

Solution:

For the hyperbola \frac{x^2}{16} - \frac{y^2}{9} = 1
we have
a² = 16,a = 4, and b² = 9,b = 3.
Foci: The foci are at (±c, 0), where
c = √a² + b² = √16 + 9 = √25 = 5
So, foci = (5, 0) and (−5, 0).
Asymptotes: The equations of asymptotes are
y = \pm \frac{b}{a} x
Substituting values:
y = \pm \frac{3}{4} x

Question 3: Find the equation of the parabola whose focus is (0,−4) and directrix is y=4.

Solution:

The standard form of a vertically oriented parabola is:
(x - h)^2 = 4a (y - k)
where (h, k) is the vertex, and a is the distance from the vertex to the focus.
The vertex lies midway between the focus and the directrix:
k = \frac{-4 + 4}{2} = 0
So, the vertex is (0,0).Distance a = ∣4−0∣=4, and since it opens downward, a=−4.
Thus, the equation is:x^2 = -16y

Question 4: Find the type of conic section represented by the equation: 4a²-9y²=36.

Solution:

Rewriting the equation in standard form: \frac{x^2}{9} - \frac{y^2}{4} = 1
This is of the form
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
which represents a hyperbola.

Unsolved Question on the Difference between Parabola and Hyperbola

Question 1: Write the standard equations of a parabola and a hyperbola and explain how their general forms differ.

Question 2: Identify whether the equation x²=8y represents a parabola or a hyperbola. Justify your answer.

Question 3: Sketch the graphs of the following equations and classify them as a parabola or hyperbola:

1. y² = 16x
2. \frac{x^2}{9} - \frac{y^2}{4} = 1

Question 4: Explain why a hyperbola has asymptotes, but a parabola does not.

Question 5:

1. The equation of a parabola is given as y² = 20x. Find its directrix and eccentricity.
2. For the hyperbola \frac{x^2}{16} - \frac{y^2}{9} = 1,find its eccentricity.

Conclusion

Parabolas and hyperbolas are both conic sections but have different shapes, properties, and applications. Parabolas are used for reflection-based systems, while hyperbolas describe motion and wave behavior. Understanding these differences helps in fields like physics, engineering, and astronomy.