Hyperbola vs Parabola is an interesting difference to study in geometry. Let us first see a few lines about these two curves and then go deep into hyperbola vs parabola.

Hyperbola is like an open curve with two branches. It is an intersection of a plane with both halves of a double cone. The plane doesn’t need to be parallel to the cone’s axis; the hyperbola will be symmetrical in any case.

A parabola is a curve, where any point is at an equal distance from a fixed point (the focus), and a fixed straight line (the directrix).

Now let us see the history of both the curves.

Index

**History**

*Menaechmus* discovered Hyperbolae during his investigations of the problem of doubling the cube but later called *sections of obtuse cones*. The term hyperbola is believed to be coined by *Apollonius of Perga* in his definitive work, the Conics, on the conic sections.

*Archimedes* computed the area enclosed by a parabola and a line segment by the method of exhaustion, 3rd century BC.

The name “parabola” by Apollonius, who discovered many properties of conic sections. It means “application,” referring to “application of areas”.

**What is a Parabola?**

Parabola is a **plane curve** which is **mirror-symmetrical** and is approximately U-shaped.

A plane curve generated by moving points so that its **distance from a fixed point is equal to its distance from a fixed-line**: the intersection of a right circular cone with a parallel plane to an element of the cone.

**General ****Equation** of Parabola

**General**of Parabola

**Equation**General equation of parabola is given by,

\(y^2 = \pm 4ax\) (directrix is parallel to y-axis), &

\(x^2 = \pm 4by\) (directrix is parallel to x-axis)

**What is a Hyperbola?**

The pair of hyperbolas are formed by the intersection of a plane and two equal right circular cones on opposite sides of the same vertex.

A hyperbola is the locus of all those points in a plane, such that the **difference in the distances from two fixed points in the plane is a constant**.

**General ****Equation of Hyperbola**

**Equation of Hyperbola**

Taken as known the focus (h, k),

The formula for the Hyperbola can be given by,

\(\frac{(x-h)^2}{a^2} – \frac{(y-k)^2}{b^2} = 1\),

Here, \(a\) is the **transverse axis** & \(b\) is the **conjugate axis**.

Let, the directrix of parabola be \(y = mx + b\), then the equation is given as

\(\frac{[y – mx – b]^2}{[m^2 + 1]} = (x – h)^2 + (y – k)^2\)

**Eccentricity**

The eccentricity is the **distance ratio from the centre to a vertex** and **from the centre to a focus(foci)**. It is denoted by \(e\).

**Eccentricity of Hyperbola** = \(\frac{c}{a}\)

**Eccentricity of hyperbola** is always \(\geq 1\).

Since, \(c \geq a\)

The eccentricity of Parabola is 1.

**Latus Rectum**

The line segments perpendicular to the transverse axis through any of the foci such that their endpoints lie on the curve are defined as the latus rectum.

The **Latus Rectum in Hyperbola** is given by \(\frac{2b^2}{a}\).

The **Latus Rectum of Parabola** is given by \(4a\).

**Applications**

The main application of parabolas is their **reflective properties** (lines parallel to the axis of symmetry reflect the focus).

Hyperbola is handy in real-world applications like telescopes, headlights, flashlights, lampshades, Gear transmission, structure for Coal-fired Power Plants, etc.

These are frequently used in **physics, engineering**, and many other areas.

**Examples**

**Question 1.** Find the points of intersection of the two parabolas with equation \(y = -(x – 3)^2 + 2\) and \(y = x^2 – 4x + 1\).

**Solution.** Given,

\(y = -(x – 3)^2 + 2 … (1)\)

and, \(y = x^2^{ } – 4x + 1 … (2)\)

Equating (1) & (2) we get,

\(-(x – 3)^2^{ } + 2 = x^2^{ } – 4x + 1\)

\(\Rightarrow – 2x^2^{ } + 10x – 8 = 0\)

\(\Rightarrow -x^2^{ } + 5 x – 4 = 0\)

Therefore \(x = 1, 4\)

Using equation to find y, we get,

\(x = 1\) in (1) \(\Rightarrow y = -(x – 3)^2^{ } + 2\)

\(\Rightarrow y= -(1 – 3)^2^{ } + 2 = – 2\)

\(x = 4\) in (2) \(\Rightarrow y = -(x – 3)^2^{ } + 2\)

\(\Rightarrow y = -(4 – 3)^2^{ } + 2 = 1\)

Therefore, the points of intersection are \((1 , -2)\) and \((4 , 1)\).

**Question 2.** Calculate the equation of the hyperbola with a transverse axis of 8 and a focal length of 10.

**Solution.** Given,

Transverse axis \((2a) = 8\) & Focal length \((2c) = 10\).

Therefore \(a = 4\) & \(c = 5\)

As we know, by the formula

\(b^2 = c^2 – a^2\)

Therefore, \(b = \sqrt{25 – 16} = 3\)

Hence, the equation of hyperbola is

\(\frac{x^2}{4^2} – \frac{y^2}{3^2} = 1\)

**FAQs**

**Is a hyperbola a parabola?**The parabola is a single open curve with eccentricity one, whereas a hyperbola has two curves with an eccentricity greater than or equal to one. Both have an open curve that extends to infinity.

**How many Directrix does hyperbola have?**Hyperbolas, as well as non circular ellipses, have two associated directrices and two distinct foci.

Each directrix being perpendicular to the line joining the two foci.

**What happens when eccentricity is infinite?**The eccentricity shows us how “un-circular” a given curve is

Circle has eccentricity = 0,

Parabola has eccentricity = 1

Hyperbola has eccentricity \(\geq\) 1

For infinite eccentricity, we get a line.

So, that was all about hyperbola vs parabola and I hope that you got to learn about both the curves and their differences