Edexcel IAL - Further Pure Mathematics 1- 4.1 Cartesian Equations of Parabolas and Hyperbolas- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 1- 4.1 Cartesian Equations of Parabolas and Hyperbolas -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 1- 4.1 Cartesian Equations of Parabolas and Hyperbolas -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 4.1 Cartesian Equations of Parabolas and Hyperbolas
Cartesian Equations of the Parabola and Rectangular Hyperbola
Conic sections such as the parabola and the rectangular hyperbola can be represented either by Cartesian equations or by parametric equations. Both forms describe the same curve but are useful in different situations.
Parabola
The standard equation of a parabola with axis along the x-axis is:
\( y^2 = 4ax \)
This parabola has:
- Vertex at the origin \( (0,0) \)
- Focus at \( (a,0) \)
- Directrix \( x = -a \)
Every point on the parabola is equidistant from the focus and the directrix.
Parametric Form of the Parabola
The parabola \( y^2 = 4ax \) can be written parametrically as:
\( x = at^2,\quad y = 2at \)
Here \( t \) is called the parameter. As \( t \) varies, the point \( (at^2, 2at) \) traces out the parabola.
Rectangular Hyperbola
A rectangular hyperbola has asymptotes that are perpendicular to each other.
The standard equation is:
\( xy = c^2 \)
This curve lies in the first and third quadrants when \( c^2 > 0 \).
The asymptotes are:
\( x = 0,\quad y = 0 \)
Parametric Form of the Rectangular Hyperbola
The hyperbola \( xy = c^2 \) can be written as:
\( x = ct,\quad y = \dfrac{c}{t} \)
Substituting gives:
\( xy = ct \cdot \dfrac{c}{t} = c^2 \)
So this parameterisation always satisfies the equation of the hyperbola.
Comparison
| Curve | Cartesian Equation | Parametric Form |
| Parabola | \( y^2 = 4ax \) | \( x = at^2,\ y = 2at \) |
| Rectangular hyperbola | \( xy = c^2 \) | \( x = ct,\ y = \dfrac{c}{t} \) |
Example
Show that the parametric equations \( x = at^2,\ y = 2at \) satisfy the parabola \( y^2 = 4ax \).
▶️ Answer / Explanation
\( y^2 = (2at)^2 = 4a^2t^2 \)
\( 4ax = 4a(at^2) = 4a^2t^2 \)
So \( y^2 = 4ax \).
Example
Find the Cartesian equation of the curve \( x = 3t,\ y = \dfrac{3}{t} \).
▶️ Answer / Explanation
\( xy = 3t \cdot \dfrac{3}{t} = 9 \)
So the equation is:
\( xy = 9 \)
Example
A point on the parabola \( y^2 = 8x \) has parameter \( t = 2 \). Find its coordinates.
▶️ Answer / Explanation
Here \( 4a = 8 \), so \( a = 2 \).
\( x = at^2 = 2 \times 4 = 8 \)
\( y = 2at = 2 \times 2 \times 2 = 8 \)
The point is \( (8, 8) \).