202212141640 Solution to 1983-CE-AMATH-II-5

Find the equation of the two lines which are both parallel to the line

$3x-2y=0$

and tangent to the ellipse

$4x^2+y^2=16$ .

Roughwork.

The slope of the line is

$\begin{aligned} 3x-2y & =0 \\ 3\,\mathrm{d}x-2\,\mathrm{d}y & = 0 \\ \textrm{Slope }m =\frac{\mathrm{d}y}{\mathrm{d}x} & = \frac{3}{2} \\ \end{aligned}$

and the gradient of the ellipse

$\begin{aligned} 4x^2+y^2 & =16 \\ 8x\,\mathrm{d}x + 2y\,\mathrm{d}y & = 0 \\ m(x,y) =\frac{\mathrm{d}y}{\mathrm{d}x} & = -\frac{4x}{y} \\ \end{aligned}$