202211221145 Solution to 1972-CE-AMATH-I-XX

The equation of the ellipse shown in the figure below is given by

$9x^2+ky^2=144$ .

Find the value of $k$ and the coordinates of the point $A$ .

Roughwork.

$\begin{aligned} 9x^2+ky^2-144 & = 0 \\ 18x\,\mathrm{d}x+2ky\,\mathrm{d}y & = 0 \\ \frac{\mathrm{d}y}{\mathrm{d}x} & = -\frac{9x}{ky} \\ \end{aligned}$

The equation of a standard ellipse centred at the origin with width $2a$ and height $2b$ is analytically

$\displaystyle{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}$ .

_{Wikipedia on Ellipse}

$\begin{aligned} 9x^2+ky^2 & = 144 \\ \frac{x^2}{(4)^2} + \frac{y^2}{(12/\sqrt{k})^2} & = 1 \\ a & = 4 \\ b & = \frac{12}{\sqrt{k}} \\ \end{aligned}$

This problem is not to be attempted.

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