Conic Sections: Parabola – Definition, Equation, Graph, Applications

TAKEAWAYS:

• A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).
• The standard equation of a parabola depends on its orientation: vertical (y² = 4ax) or horizontal (x² = 4ay).
• Key features include the vertex (turning point), focus (focal point), and directrix (axis line).
• Parabolas can be graphed by plotting points using the equation and identifying symmetry.
• Applications include satellite dishes, headlights, and parabolic reflectors.

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1. What is a Parabola?

A parabola is a conic section formed when a plane intersects a double cone at an angle parallel to one of its sides. It is a U-shaped curve that is symmetric about its axis.

Key Terms:

• Focus (F): A fixed point inside the parabola.
• Directrix: A fixed line outside the parabola.
• Vertex (V): The midpoint of the focus and directrix; the "tip" of the parabola.
• Axis of Symmetry: The line passing through the focus and vertex.

Definition:

A parabola is the locus of a point that moves such that its distance from the focus is always equal to its distance from the directrix.

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2. Standard Equations of a Parabola

The equation of a parabola depends on its orientation (whether it opens up/down or left/right).

Case 1: Vertical Parabola (Opens Up or Down)

• Equation: $y^2=4ax$
  • Opens to the right if $a>0$.
  • Opens to the left if $a<0$.
• Focus: $(a,0)$
• Directrix: $x=-a$
• Vertex: $(0,0)$

Case 2: Horizontal Parabola (Opens Left or Right)

• Equation: $x^2=4ay$
  • Opens upwards if $a>0$.
  • Opens downwards if $a<0$.
• Focus: $(0,a)$
• Directrix: $y=-a$
• Vertex: $(0,0)$

Case 3: Parabola with Vertex at (h, k)

If the vertex is not at the origin, the equations become:

• Vertical: $(y-k{)}^2=4a(x-h)$
• Horizontal: $(x-h{)}^2=4a(y-k)$

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3. Deriving the Standard Equation

Let’s derive $y^2=4ax$ for a vertical parabola.

1. Let $F=(a,0)$ be the focus and the directrix be $x=-a$.
2. Let $P=(x,y)$ be any point on the parabola.
3. By definition, distance from $P$ to $F$ = distance from $P$ to the directrix. $$\sqrt{(x-a{)}^2+y^2}=|x+a|$$
4. Square both sides: $$(x-a{)}^2+y^2=(x+a{)}^2$$
5. Expand and simplify: $$x^2-2ax+a^2+y^2=x^2+2ax+a^2$$ $$y^2=4ax$$

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4. Graphing a Parabola

To graph $y^2=4ax$:

1. Plot the vertex at $(0,0)$.
2. Plot the focus at $(a,0)$.
3. Draw the directrix as $x=-a$.
4. Choose points for $x$ and solve for $y$:
   • If $x=1$, $y^2=4a(1)\Rightarrow y=\pm 2\sqrt{a}$.
   • If $x=4$, $y^2=16a\Rightarrow y=\pm 4\sqrt{a}$.
5. Plot these points and draw a smooth curve.

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5. Latus Rectum

The latus rectum is the line segment perpendicular to the axis of symmetry through the focus, with endpoints on the parabola.

• Length of latus rectum: $|4a|$.

For $y^2=4ax$, the latus rectum is the line $x=a$, and its length is $4a$.

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6. Applications of Parabolas

Parabolas are used in:

1. Satellite Dishes: Reflect signals to a single point (focus).
2. Headlights: Reflect light in a parallel beam.
3. Bridges: Parabolic arches distribute weight efficiently.
4. Projectile Motion: Paths of thrown objects follow a parabola.

A satellite dish uses a parabolic reflector to focus signals. (Image: Public domain, via Wikimedia Commons)

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7. Solved Examples

Example 1: Find the Focus and Directrix

Given the equation $y^2=8x$, find the focus and directrix.

Solution:

1. Compare with $y^2=4ax$: $$4a=8\Rightarrow a=2$$
2. Focus: $(a,0)=(2,0)$
3. Directrix: $x=-a=-2$

Answer: Focus = $(2,0)$, Directrix = $x=-2$.

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Example 2: Find the Equation of a Parabola

Find the equation of a parabola with vertex at $(0,0)$ and focus at $(0,3)$.

Solution:

1. Since the focus is on the y-axis, the parabola opens upwards.
2. Use the standard form $x^2=4ay$.
3. Here, $a=3$, so: $$x^2=4(3)y\Rightarrow x^2=12y$$

Answer: The equation is $x^2=12y$.

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Example 3: Find the Vertex and Focus

Given $(y-2{)}^2=12(x+1)$, find the vertex and focus.

Solution:

1. Compare with $(y-k{)}^2=4a(x-h)$: $$k=2,h=-1,4a=12\Rightarrow a=3$$
2. Vertex: $(h,k)=(-1,2)$
3. Focus: $(h+a,k)=(-1+3,2)=(2,2)$

Answer: Vertex = $(-1,2)$, Focus = $(2,2)$.

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8. Comparison Table: Vertical vs. Horizontal Parabolas

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9. NEB Board-Style Questions

Short Answer Questions

1. Define a parabola. What are its key features?
2. Write the standard equation of a parabola that opens to the right with vertex at $(0,0)$.
3. If the focus of a parabola is $(3,0)$ and the directrix is $x=-3$, find its equation.
4. What is the length of the latus rectum for $y^2=16x$?

Long Answer Questions

5. Derive the standard equation of a parabola $y^2=4ax$.
6. A parabola has its vertex at $(2,-1)$ and focus at $(2,3)$. Find its equation.
7. Explain the applications of parabolas in real life with examples.

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Exam Tip

• Memorize standard forms: $y^2=4ax$ (horizontal) and $x^2=4ay$ (vertical).
• Identify orientation: If the squared term is $y$, it opens left/right; if $x$, it opens up/down.
• Practice graphing: Always plot the vertex, focus, and directrix first.
• Watch for shifts: If the equation has $(x-h)$ or $(y-k)$, the vertex is not at the origin.
• Applications: Know how parabolas are used in satellite dishes, headlights, and bridges.

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End of Note