Level 3 Differentiation Walkthrough

2023 NCEA Level 3 Differentiation Question 1(d)

2023 Paper

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Question

The curve is given parametrically by

\[ x=4\cos\theta \] \[ y=4\sin\theta \]

A tangent passes through the point \(P(p,q)\) on the circle.

Show that the equation of the tangent line is

\[ px+qy=p^2+q^2. \]

First walkthrough idea

Hint to try first

Differentiate both parametric equations with respect to \(\theta\).

Step 1

Find the parametric gradient

Since \(x=4\cos\theta\) and \(y=4\sin\theta\), the gradient simplifies neatly to \(-\frac{x}{y}\).

Show the first step’s working
\[ \frac{dx}{d\theta}=-4\sin\theta \qquad \frac{dy}{d\theta}=4\cos\theta \]

Since \(x=4\cos\theta\) and \(y=4\sin\theta\), the gradient simplifies neatly to \(-\frac{x}{y}\).

Key result

\[ \frac{dy}{dx}=-\frac{x}{y} \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Parametric differentiation and building a tangent equation.

This is Question 1(d) from the 2023 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise parametric differentiation and building a tangent equation. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Use the derivative for the gradient and the original curve for the point before forming the tangent equation.

Continue practising

← Question 1(c) Question 1(e) →