Standard Equations and Basic Properties

Parabola: Standard Equations and Basic Properties

Welcome to the world of parabolas! This isn't just another conic section; it's a fundamental shape with applications ranging from satellite dishes to projectile motion, making it a crucial topic for JEE Main. Mastering parabolas will not only boost your coordinate geometry skills but also enhance your problem-solving abilities across various physics and math topics.

What is a Parabola?

Imagine a point (the focus) and a line (the directrix). A parabola is the set of all points that are equidistant from the focus and the directrix. Think of it as a curved line that "hugs" the focus while staying away from the directrix. This definition gives the parabola its unique U-shape.

Geometric Interpretation: Visualise folding a piece of paper so that a fixed point (focus) lands on a fixed line (directrix). The crease forms a parabola! This helps to intuitively grasp the definition.

Standard Forms of the Parabola

The equation of a parabola simplifies beautifully when we place the focus and directrix in specific locations. Here are the four standard forms you absolutely must know:

1. y² = 4ax

   This is the most common standard form. The parabola opens to the right. Let's break down its properties:

   • Vertex: The vertex is at the origin, $(0, 0)$. It's the "tip" of the parabola.
   • Focus: The focus is at $(a, 0)$. Notice that 'a' determines the distance of the focus from the vertex.
   • Directrix: The directrix is the line $x = -a$. It's a vertical line 'a' units to the left of the vertex.
   • Axis: The axis is the line $y = 0$ (the x-axis). It's the line of symmetry of the parabola.
   • Latus Rectum: The latus rectum is a line segment through the focus, perpendicular to the axis, with endpoints on the parabola. Its length is $4a$.
   $$y^2 = 4ax \text{ (focus: } (a,0), \text{ directrix: } x = -a \text{)}$$

   Example: Consider $y^2 = 8x$. Here, $4a = 8$, so $a = 2$. The focus is at $(2, 0)$ and the directrix is $x = -2$.

2. y² = -4ax

   This parabola opens to the left. The focus is at $(-a, 0)$ and the directrix is $x = a$. Everything else is similar to the $y^2 = 4ax$ case.

3. x² = 4ay

   Now the parabola opens upwards. Here's what changes:

   • Vertex: Still at the origin, $(0, 0)$.
   • Focus: The focus is now at $(0, a)$.
   • Directrix: The directrix is the line $y = -a$. It's a horizontal line 'a' units below the vertex.
   • Axis: The axis is the line $x = 0$ (the y-axis).
   • Latus Rectum: The latus rectum is still $4a$.
   $$x^2 = 4ay \text{ (focus: } (0,a), \text{ directrix: } y = -a \text{)}$$

   Example: For $x^2 = 12y$, we have $4a = 12$, so $a = 3$. The focus is at $(0, 3)$ and the directrix is $y = -3$.

4. x² = -4ay

   This parabola opens downwards. The focus is at $(0, -a)$ and the directrix is $y = a$.

Parametric Representation

Sometimes, it's easier to describe points on a parabola using a parameter, typically denoted by 't'. For the standard parabola $y^2 = 4ax$, the parametric representation is:

This means any point $(x, y)$ on the parabola can be written as $(at^2, 2at)$ for some value of 't'. 't' is just a parameter that helps us generate points on the curve.

Geometric Interpretation: As 't' varies, the point $(at^2, 2at)$ traces out the parabola. Different values of 't' correspond to different locations on the curve.

Latus Rectum

We briefly mentioned the latus rectum. It's the line segment passing through the focus, perpendicular to the axis, with endpoints on the parabola. Its length is always $4a$. The endpoints of the latus rectum are $(a, 2a)$ and $(a, -2a)$ for $y^2 = 4ax$. Knowing the latus rectum helps visualize the "width" of the parabola.

Eccentricity

Eccentricity is a measure of how "stretched" a conic section is. For a parabola, the eccentricity is always 1. This distinguishes it from ellipses (e < 1) and hyperbolas (e > 1).

Shifted Parabola Equations

What if the vertex isn't at the origin? We can shift the parabola horizontally and vertically. If the vertex is at $(h, k)$, the equation becomes:

• $(y - k)^2 = 4a(x - h)$ (opens right)
• $(x - h)^2 = 4a(y - k)$ (opens upward)

To analyze these shifted parabolas, remember these simple rules:

• Replace $x$ with $(x - h)$ and $y$ with $(y - k)$ in the standard equations.
• The focus shifts to $(h + a, k)$ for $(y - k)^2 = 4a(x - h)$.
• The directrix becomes $x = h - a$ for $(y - k)^2 = 4a(x - h)$.

Example: Consider $(y - 2)^2 = 4(x + 1)$. Here, the vertex is $(-1, 2)$ and $a = 1$. The focus is at $(-1 + 1, 2) = (0, 2)$, and the directrix is $x = -1 - 1 = -2$.