2023 ONE(d)

Merit
Question
The diagram below shows a tangent passing through the point P (p,q)(p,q) which lies on the circle
with parametric equations x=4cos⁡θx = 4\cos\theta and y=4sin⁡θy = 4\sin\theta.
Loading diagram...
Show that the equation of the tangent line is px+qy=p2+q2px + qy = p^2 + q^2.
Official Answer
x=4cos⁡θx=4\cos\theta and y=4sin⁡θy=4\sin\theta
dxdθ=−4sin⁡θ\dfrac{dx}{d\theta}=-4\sin\theta and dydθ=4cos⁡θ\dfrac{dy}{d\theta}=4\cos\theta
dydx=4cos⁡θ−4sin⁡θ=−xy=−pq\dfrac{dy}{dx}=\dfrac{4\cos\theta}{-4\sin\theta}=-\dfrac{x}{y}=-\dfrac{p}{q}
OR
Equation of circle: x2+y2=16x^2+y^2=16
Differentiating implicitly gives
2x+2ydydx=02x+2y\dfrac{dy}{dx}=0
dydx=−xy\dfrac{dy}{dx}=-\dfrac{x}{y}
At (p,q)(p,q), the gradient is −pq-\dfrac{p}{q}
Equation of tangent:
y−q=−pq(x−p)y-q=-\dfrac{p}{q}(x-p)
qy−q2=−px+p2qy-q^2=-px+p^2
px+qy=p2+q2px+qy=p^2+q^2 as required.
Grading Criteria

Achievement (u)

  • Correct derivative for dydx\dfrac{dy}{dx}.
  • dydx\dfrac{dy}{dx} can be expressed in terms of θ\theta or x,yx,y or p,qp,q).

Merit (r)

  • Proof completed, with correct .

Excellence T1

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Excellence T2

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Video Explanation
2023 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 12:26)
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Exam Paper (2023)Assessment Schedule (2023)
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