Parabola

The parabola, a conic section, is a symmetric U-shaped curve. It appears in everyday scenarios and physics, showcasing its versatility. The parabola's mirror symmetry is evident from the trajectory of a thrown ball to the design of satellite dishes. Its ubiquity underscores its significance across diverse applications, unifying various phenomena under a single mathematical framework.

The parabola, integral to conic sections, arises from slicing a right circular cone. Defined by quadratic equations, its graph depicts points that are equal in distance from a fixed line called the directrix and point (the focus). This elegant curve embodies fundamental geometric principles, with applications ranging from projectile motion to satellite dish design. 

1.0Parabola Definition

A parabola is a curve formed by all points in a plane that are equidistant from a specified point (referred to as the focus) and a fixed line (known as the directrix). 

• The fixed point is called the Focus of the Parabola.
• The fixed straight line is called the Directrix.
• A line passing through the focus and is perpendicular to the directrix is termed the axis of symmetry of the parabola.
• The point where the parabola intersects its axis is known as the vertex of the parabola.

2.0Parabola General Equation

The general equation of a parabola with axis parallel to y-axis form can be expressed as:

y = ax² + bx + c

Where:

• a symbolizes the coefficient of the quadratic term, influencing the direction and “width” of the parabola.
• b symbolizes the coefficient of the linear term, affecting the horizontal shift of the parabola.
• c symbolizes the constant term, determining the vertical shift of the parabola.

This equation allows us to describe various types of parabolas, including those that open upwards or downwards and may be shifted horizontally or vertically.

3.0Standard Equations of Parabola

The equation of a parabola is simplest when the vertex is located at the origin and the axis of symmetry aligns with either the x-axis or the y-axis. The 4 possible orientations of the parabola are shown below.

1. y² = 4ax 
2. y² = –4ax
3. x² = 4ay
4. x² = –4ay

These 4 equations are known as standard equations of parabolas.

Observations from these equations

• A parabola's symmetry with respect to its axis depends on the term in its equation. If the equation includes a y² term, the axis of symmetry aligns with the x-axis. If it contains an x² term, the axis of symmetry aligns with the y-axis.
• When the axis of symmetry coincides with the x-axis, the parabola opens in the following manner:

1. Toward the right, if the coefficient of x is positive.
2. Toward the left, if the coefficient of x is negative.

• When the axis of symmetry coincides with the y-axis, the parabola opens in the following manner:

3. Upwards when the coefficient of y is positive.
4. Downwards when the coefficient of y is negative.

4.0Latus Rectum of Parabola

The parabola latus rectum is a line segment that passes through its focus and is perpendicular to its axis. It has its endpoints on the parabola and is parallel to the directrix. The length of the latus rectum is typically denoted by 4a, where a is the distance from the vertex to the focus.

5.0Eccentricity of parabola

Parabola eccentricity is defined as the ratio of the distance from the focus to any point on the parabola to the perpendicular distance from that point to the directrix.

For a parabola, the eccentricity is always equal to 1. This means that the distance from any point on the parabola to the focus is always equal to the perpendicular distance from that point to the directrix. Unlike ellipses and hyperbolas, where the eccentricity can vary.

$\frac{\mathrm{PF}}{\mathrm{PM}}=\text{ constant }=\mathrm{e}$

constant= e[/latex]“e” is the eccentricity, which is equal to 1.

6.0Chord Joining Two Points

Parametric equation of a chord joining two points “t₁” and “t₂” of the parabola y² = 4ax is

2x– y (t₁ + t₂) + 2at₁t₂ = 0 

7.0Equation of Tangent to the Parabola

A tangent to a parabola is a straight line that touches the curve at a single point.

Slope Form

Equation of Tangent to Parabola y² =4ax for all m ≠ 0 at (a/m², 2a/m) is: y = mx + a/m

Cartesian Form

Equation of Tangent to Parabola y² =4ax at point (x₁, y₁) is: yy₁ = 2a (x + x₁)

Parametric Form

Equation of Tangent to Parabola y² =4ax at the point P (at², 2at) is: ty = x + at²

8.0Equation of Normal to the Parabola

The normal to a parabola is a line perpendicular to the tangent at a given point and passing through that point.

Slope Form

Equation of Normal to Parabola y² =4ax for all m ≠ 0 at (am², –2am) is: y = mx – 2am – am³.

Cartesian Form

Equation of Normal to Parabola y² =4ax at point (x₁, y₁) is: y – y₁ = ( –y₁/2a) (x – x₁)    

Parametric Form

Equation of Normal to Parabola y² =4ax at point (at², 2at) is: y + tx = 2at + at³.

9.0Parabola Solved Examples

Example 1: Find the vertex, axis, directrix, and the length of the latus rectum for the given parabola: y² =12x

Solution: The given equation is in the form y² =4ax, where 4a = 12, a = 3.

Vertex = (0,0)

Axis = y = 0

Directrix = x = –a So, x = –3

Length of Latus Rectum ⇒ 4a = (4 × 3) units = 12 units. 

Example 2: Find the axis, vertex, tangent at the vertex, directrix, and the length of the latus rectum of the given parabola: y² + 4y – 2x – 2 = 0

Solution: The given solution can be re-written as (y + 2)² = 2(x + 3)

Which is in the form Y² =4aX

where Y = y + 2, X = x + 3, 4a = 2

Hence, the axis ⇒ Y = 0 ⇒ y + 2 = 0 ⇒ y = –2

 The vertex is X = 0, Y = 0, i.e. (– 3, –2).

The tangent at the vertex is:

X = 0 ⇒ x + 3 = 0 ⇒ x = – 3

The directrix is X + a = 0.

⇒ x + 3 + ½ = 0 

⇒ x = – 7/2

Length of the latus rectum = 4a = 2 units.

Example 3:  Determine the equation of the parabola whose focus is ( –2, 3) and a directrix given by the line x + y + 2 = 0.

Solution: Let P(x, y) be any point on the parabola. then 

$\Rightarrow \sqrt{(x+2{)}^2+(y-3{)}^2}=|x+y+2|/\sqrt{1+1}$

⇒ (x +2)² +(y – 3)² = (x +y  + 2)²/2

⇒ 2( x² + 4x + 4 + y² – 6y + 9) = x² +y² + 4 + 4x + 4y + 2xy

⇒ x² + y² + 4x – 16y + 2xy + 22 = 0

Example 4: How do you find the vertex of a parabola?

Ans: The parabola vertex can be found using the formula $\left(-\frac{\mathrm{b}}{2\mathrm{a}},\frac{4\mathrm{ac}-{\mathrm{b}}^2}{4\mathrm{a}}\right)$ for a parabola in standard form y=ax² + bx + c.