N2021 P1 Q3

N2021 P1 Q3

Junior College 2
6 marks
Free

A curve has equation \(x^{\frac12}+y^{\frac12}=3\), for \(x>0\) and \(y>0\).

  1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\left(\frac{y}{x}\right)^{\frac12}\).[2]
  2. Find the equation of the normal to the curve at the point where \(x=1\).[4]

Default solution

  1. \(\frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}\frac{\mathrm{d}y}{\mathrm{d}x}=0\)
    \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{\sqrt{y}}{\sqrt{x}}=-\left(\frac{y}{x}\right)^{\frac12}\) (shown)
  2. \(x=1:\quad \sqrt{y}=2\Rightarrow y=4\)
    \(m_{\mathrm{t}}=-2;\quad m_{\mathrm{n}}=\frac12\)
    \(y-4=\frac12(x-1)\)
    \(\therefore y=\frac12x+\frac72\)
Answer:(a) \(\frac{\mathrm{d}y}{\mathrm{d}x}=-\sqrt{\frac{y}{x}}\); (b) \(y=\frac12x+\frac72\).

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