1. Discuss each of the following conic sections and sketch the graph: (a) \( 63 y^{2}+42 y-18 x^{2}+36 x \sqrt{2}=155 \) (b) \( 25 y^{2}+100 y-4 x^{2}-48 x-44=0 \) 2. A conic section with centre at \( (\sqrt{2},-1) \) has the value of eccentricity given by \( e=\frac{3 \sqrt{2}}{2} \). Given that the focal axis is horizontal and that the distance between foci and directrices is seven thirds, (a) write down the conic section's equation (b) write down the equations of directrices. 3. The opening of a tunnel is in the shape of a semi-elliptical arch. The arch is 18 metres wide and 12 metres high. Find the height of the arch 4 metres from the edge of the tunnel.
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The Deep Dive
To dive into the conic sections you mentioned, we need to analyze and rearrange the equations you've provided. Conic sections have intriguing histories and numerous applications, so let's consider some engaging insights on them! In the case of the first equation \( 63 y^{2}+42 y-18 x^{2}+36 x \sqrt{2}=155 \), with some tidying up, we see that this is a hyperbola. As for the second equation \( 25 y^{2}+100 y-4 x^{2}-48 x-44=0 \), when simplified, showcases a different conic—generally a hyperbola as well, depending on the standard form. Did you know that conic sections were studied extensively by ancient Greek mathematicians? They were originally classified based on their shapes, projecting fascinating insights into geometric properties that have influenced various fields including astronomy and engineering! Now, moving on to practical applications, conic sections like ellipses and parabolas underlie the design of orbits in astrophysics, the paths of projectiles in engineering, and even the data patterns in economics! For instance, the parabolic shape of satellite dishes is crucial as it allows for the optimal collection of signals. Isn't it amazing how these mathematical concepts shape our world around us? Let's sketch those graphs and dive into their properties while having fun with some real-world applications!
