Parabola

A parabola is the curve formed by all the points that are equally far from one point (the focus) and one line (the directrix). This single 'equal distance' rule is what creates every property of the parabola. It is also why satellite dishes and car headlights are shaped like parabolas: they gather light at the focus and send it back out in parallel rays. Drag the focal distance p slider to watch the parabola widen and narrow, and to confirm that the distance from a point P to the focus always equals its distance to the directrix.

Intuition of Focus and Directrix

Focal Distance p1.5

👀 What is a Parabola?

①Pick any point P on the parabola

②The distance to focus F equals the distance to the directrix

③This 'equal distance' condition defines a parabola

④Light from the focus reflects off the parabola in parallel rays — principle of satellite dishes and headlights!

Shape Change with p

p Value1.5

📏 Meaning of p

①Large p: focus is far, parabola spreads wide

②Small p: focus is near, parabola is narrow

③The directrix is always at equal distance on the opposite side of the focus

Deriving the Standard Form

Standard Form (Horizontal)

y² = 4px

Focus F(p, 0), directrix x = −p, vertex at origin

Standard Form (Vertical)

x² = 4py

Focus F(0, p), directrix y = −p, vertex at origin

🔍 Derivation Steps

①Definition: PF = Pd (distance to focus = distance to directrix)

②From P(x,y) to F(p,0): √((x−p)² + y²)

③Distance to directrix x=−p: |x+p|

④Squaring both sides: (x−p)² + y² = (x+p)²

⑤Expanding gives y² = 4px

When Vertex Is Not at Origin

Translated Parabola

(y − k)² = 4p(x − h)

Vertex (h, k), focus (h+p, k)

🎯 Translation Essentials

①Replace x with (x−h), y with (y−k)

②Focus shifts by (h,k) in the same direction

③Directrix becomes x = h−p

④Formula structure is unchanged

Work It Out

Example 1

Find the focus and directrix of the parabola y²=12x.

Compare with the standard form y²=4px to find p.

4p=12 ⇒ p=3

The focus is (p, 0) and the directrix is x=−p.

focus (3, 0), directrix x=−3

▸ focus (3, 0), directrix x=−3

In the form y²=4px, comparing the value of 4p gives the focus and directrix at once.

Example 2

Find the coordinates of the focus of the parabola (y−1)²=8(x−2).

The vertex is (2, 1) and from 4p=8 we get p=2.

4p=8 ⇒ p=2, vertex (2, 1)

The focus is at (h+p, k), shifted from the vertex by p along x.

focus (2+2, 1) = (4, 1)

▸ (4, 1)

For a translated parabola, add p to the vertex (h,k) to get the focus. The directrix is x=h−p=0.

Summary

Key Formula

y² = 4px

Focus (p,0) | Directrix x=−p | p>0: opens right, p<0: opens left

2023 KICE mock exam Math (Geometry) type, adapted

For the parabola y²=4px with focus (2, 0), what is the equation of the directrix?

①x=−2

②x=2

③x=−1

④x=1

⑤y=−2

▸ ① x=−2

The focus of y²=4px is (p, 0), so p=2.

The directrix is x=−p.

directrix x=−2

🎯 Exam Points

①Definition: distance to focus = distance to directrix

②In y²=4px, the sign of 4p determines opening direction

③Practice quickly finding focus and directrix equations

④For vertex (h,k), use translation formula

⑤Parabolic focus connects to dish antenna and headlight principles

Ellipse

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