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A designer writes the curve 4x² + 9y² - 16x + 18y - 11 = 0 on a plan whose lengths are in metres…

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A designer writes the curve $4x^2 + 9y^2 - 16x + 18y - 11 = 0$ on a plan whose lengths are in metres. Work out $h^2 - ab$ for it, using $a = 4$ , $b = 9$ and $h = 0$ . Then find the coordinates of its centre, and the distance of a focus from the centre.

Need a reminder? The key ideaShow

The tangent at $(x_1, y_1)$ is $\dfrac{xx_1}{a^2} \pm \dfrac{yy_1}{b^2} = 1$ . With slope $m$ : $c^2 = a^2m^2 \pm b^2$ . The director circle is $x^2 + y^2 = a^2 \pm b^2$ . For $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ the sign of $h^2 - ab$ names the conic when $\Delta \ne 0$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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