Coordinate Systems (FP1)

Edexcel International A-Level Further Mathematics· FP1 4.1 to 4.4 (2018 spec Issue 3)· 25 min read

1. Standard Forms of Parabolas and Rectangular Hyperbolas★★☆☆☆⏱ 5 min

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📘 Definition

Standard Conic Forms for FP1

The only two conic sections examinable in FP1 coordinate systems are the parabola and rectangular hyperbola, with fixed standard and parametric forms given in the formula book.

Example:

Parabola: Cartesian $y^{2} = 4 a x$ , parametric $(a t^{2}, 2 a t)$ ; Rectangular hyperbola: Cartesian $x y = c^{2}$ , parametric $(c t, \frac{c}{t})$ .

📐 Worked Example

State the coordinates of the point on the parabola $y^{2} = 12 x$ corresponding to parameter t=2, and write the Cartesian equation of a rectangular hyperbola with c=4.

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Step 1: For the parabola, compare $y^{2} = 12 x$ to standard $y^{2} = 4 a x$ to find $4 a = 12$ , so $a = 3$ .
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Step 2: Use parametric coordinates for parabola: $(a t^{2}, 2 a t) = (3 \times 2^{2}, 2 \times 3 \times 2) = (12, 12)$ .
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Step 3: For the rectangular hyperbola, substitute $c = 4$ into $x y = c^{2}$ : $x y = 16$ .
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Final answer: Parabola point = (12,12), hyperbola equation $x y = 16$ .

Note that the parameter $t$ is only used to define general points on the curves in FP1: you will not be required to manipulate parametric equations beyond substituting values or using them to define points for tangent/normal calculations.

2. Focus-Directrix Property of the Parabola★★★☆☆⏱ 7 min

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📘 Definition

Focus and Directrix of $y^2=4ax$

The parabola $y^{2} = 4 a x$ has a focus at point $(a, 0)$ and a directrix line given by $x = - a$ . All points on the parabola are equidistant from this focus and directrix.

Example:

For $y^{2} = 8 x$ , $a = 2$ , so focus is (2,0) and directrix is $x = - 2$ .

📐 Worked Example

A point P lies on the parabola $y^{2} = 20 x$ . Show that the distance from P to the focus equals its perpendicular distance to the directrix for the point P with y-coordinate 10.

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Step 1: Find $a$ for $y^{2} = 20 x$ : $4 a = 20$ , so $a = 5$ . Focus is (5, 0), directrix $x = - 5$ .
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Step 2: Find coordinates of P: substitute $y = 10$ into $y^{2} = 20 x$ : $100 = 20 x \to x = 5$ , so $P = (5, 10)$ .
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Step 3: Calculate distance from P to focus: $(5 - 5)^{2} + (10 - 0)^{2} = 100 = 10$ .
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Step 4: Calculate perpendicular distance from P to directrix $x = - 5$ : horizontal distance since directrix is vertical: $5 - (- 5) = 10$ .
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Step 5: Both distances equal 10, so the property holds for point P.

3. Tangents to Parabolas and Rectangular Hyperbolas★★★☆☆⏱ 7 min

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To find the equation of a tangent to either curve, you must differentiate the Cartesian form of the curve to find the gradient of the tangent at the given point, then use the straight line equation $y - y_{1} = m (x - x_{1})$ . Remember parametric differentiation is not permitted here.

📐 Worked Example

Find the equation of the tangent to the rectangular hyperbola $x y = 9$ at the point where $x = 3$ . Give your answer in the form $a x + b y + c = 0$ where $a, b, c$ are integers.

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Step 1: Rearrange hyperbola equation to Cartesian form for differentiation: $y = \frac{9}{x} = 9 x^{- 1}$ .
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$\frac{dy}{dx} = -9x^{-2} = -\frac{9}{x^2}$
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Step 2: Find coordinates of the point: when $x = 3$ , $y = 9/3 = 3$ , so point is (3,3).
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Step 3: Calculate tangent gradient at $x = 3$ : $m = - \frac{9}{3 ^{2}} = - 1$ .
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Step 4: Substitute into straight line equation: $y - 3 = - 1 (x - 3)$
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Step 5: Rearrange to required form: $y - 3 = - x + 3 \to x + y - 6 = 0$ .

4. Normals to Parabolas and Rectangular Hyperbolas★★★★☆⏱ 6 min

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The normal to a curve at a point is perpendicular to the tangent at that point, so its gradient is the negative reciprocal of the tangent gradient. Follow the same process as for tangents, but swap the gradient to $m_{n or ma l} = - 1/ m_{t an g e n t}$ after calculating the tangent gradient.

📐 Worked Example

Find the equation of the normal to the parabola $y^{2} = 16 x$ at the point (1, 4). Give your answer in the form $y = m x + c$ .

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Step 1: Rearrange parabola equation to Cartesian form for differentiation (take positive root as $y = 4$ is positive): $y = 4 x = 4 x^{1/2}$ .
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$\frac{dy}{dx} = 4 \times \frac{1}{2}x^{-1/2} = \frac{2}{\sqrt{x}}$
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Step 2: Calculate tangent gradient at $x = 1$ : $m_{t an g e n t} = 2/ 1 = 2$ .
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Step 3: Calculate normal gradient: $m_{n or ma l} = - 1/2$ .
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Step 4: Substitute point (1,4) and normal gradient into straight line equation: $y - 4 = - \frac{1}{2} (x - 1)$
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Step 5: Rearrange to $y = m x + c$ : $y = - \frac{1}{2} x + \frac{1}{2} + 4 = - \frac{1}{2} x + \frac{9}{2}$ .

5. Common Pitfalls

Wrong move:

Using parametric differentiation to find tangent/normal gradients

Why:

Edexcel FP1 explicitly forbids parametric differentiation for this topic, and you will lose method marks even if your final answer is correct

Correct move:

Always differentiate the Cartesian form of the curve ( $y$ as a function of $x$ ) to calculate gradients

Wrong move:

Using focus and directrix of the rectangular hyperbola to solve problems

Why:

These are explicitly marked as non-examinable in the FP1 formula book, and questions will never require them for FP1 assessments

Correct move:

Only use the focus-directrix property for the parabola $y^{2} = 4 a x$ in FP1 coordinate systems questions

Wrong move:

Taking the wrong sign of the square root when rearranging $y^{2} = 4 a x$ for differentiation

Why:

This will give you the negative of the correct tangent gradient, leading to incorrect tangent and normal equations

Correct move:

Use the sign of the y-coordinate of the point you are working with when taking the square root of $4 a x$

Wrong move:

Using general hyperbola $x^{2} / a^{2} - y^{2} / b^{2} = 1$ or ellipse equations in answers

Why:

These are part of FP3 content, not FP1, and will not be accepted as valid answers for FP1 coordinate systems questions

Correct move:

Only use standard forms $y^{2} = 4 a x$ for parabolas and $x y = c^{2}$ for rectangular hyperbolas in FP1

Wrong move:

Forgetting that the normal gradient is the negative reciprocal of the tangent gradient

Why:

This will result in a line parallel to the tangent instead of perpendicular, costing you accuracy marks

Correct move:

Always calculate $m_{n or ma l} = - 1/ m_{t an g e n t}$ before substituting into the straight line equation for normals

6. Quick Reference Cheatsheet

Concept	Standard Form / Formula	Notes
Parabola Cartesian	$y^{2} = 4 a x$	$a > 0$ , opens to the right
Parabola Parametric	$(a t^{2}, 2 a t)$	$t$ is any real number
Parabola Focus	$(a, 0)$	Given in FP1 formula book
Parabola Directrix	$x = - a$	Given in FP1 formula book
Rectangular Hyperbola Cartesian	$x y = c^{2}$	$c > 0$ , asymptotes at $x = 0$ , $y = 0$
Rectangular Hyperbola Parametric	$(c t, c / t)$	$t \neq = 0$
Tangent Gradient	$m_{t an} = \frac{d y}{d x}$ (Cartesian form)	No parametric differentiation allowed
Normal Gradient	$m_{n or m} = - 1/ m_{t an}$	Perpendicular to tangent

7. Frequently Asked

Do I need to use parametric differentiation to find tangents/normals in FP1?

No, Edexcel explicitly forbids parametric differentiation for this FP1 topic. You must differentiate the Cartesian form of the curve (e.g. $y = 2 a x$ for parabolas, $y = \frac{c ^{2}}{x}$ for rectangular hyperbolas) to find the gradient of the tangent, even if parametric differentiation would give the same result.

Do I need to memorise the focus and directrix of the rectangular hyperbola for FP1?

No, the FP1 formula book explicitly states these are not required for FP1 assessment. Only the focus and directrix of the parabola $y^{2} = 4 a x$ are examinable in this topic.

Going deeper

formula_bookEdexcel IAL Further Maths FP1 Formula BookReference section for conic standard forms

What's Next

Mastering FP1 coordinate systems is a foundational skill for further conic sections content in FP3, where you will explore general hyperbolas, ellipses, and their eccentricity properties. It also pairs closely with FP1 locus problems, which often ask you to find the path of intersections of tangents and normals to these curves. Before moving on, make sure you can consistently solve full exam-style questions involving tangents, normals, and the focus-directrix property, as these are high-frequency 5-7 mark questions in every FP1 paper. You should also practice combining this content with algebraic manipulation skills from P1 and P2 to simplify equations of lines and solve for intersection points efficiently.