Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on real-world applications of conic sections:

• Circle — signal range, locus of equidistant points
• Parabola — reflective dishes, projectile trajectories
• Ellipse — planetary orbits, whispering galleries
• Hyperbola — navigation systems (LORAN), cooling tower cross-sections
• Geometric properties: foci, directrices, vertices, and eccentricity
• Locus problems and coordinate geometry

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Important Formulas

Circle

$(x-h{)}^2+(y-k{)}^2=r^2$

A point $(x_1,y_1)$ is within signal range $r$ of a tower at $(h,k)$ if: $\sqrt{(x_1-h{)}^2+(y_1-k{)}^2}\le r$

Parabola (Standard Forms)

$y^2=4ax\text{(opens right)}{x}^2=4ay\text{(opens up)}$

For a parabolic dish with vertex at the origin, depth $c$, and half-diameter $d/2$, the focus (microphone position) is found by substituting $(c,d/2)$ into $y^2=4ax$ and solving for $a$.

Ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\text{where}{c}^2=a^2-b^2,e=\frac{c}{a}<1$

Orbital distances (Sun at one focus):

• Perihelion (minimum distance) $=a-c$
• Aphelion (maximum distance) $=a+c$

Hyperbola

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{where}{c}^2=a^2+b^2,e=\frac{c}{a}>1$

Navigation (LORAN): The constant difference in signal arrival times gives: $2a=v\times \Delta t$ where $v$ is signal speed and $\Delta t$ is the time delay.

Hyperbolic cross-section width: For a pillar with equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and total height $H$, substitute $y=H/2$ and solve for $x$; the width at the top is $2x$.

Circumference and Linear Speed: $C=2\pi r$ $\text{Linear speed}=\frac{2\pi r\times n}{t}$ where $n$ = number of revolutions and $t$ = time.

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Summary

This exercise demonstrates practical applications of conic sections across physics, engineering, and architecture:

Key skills: translating physical constraints into standard conic equations, using eccentricity to relate parameters, and applying geometric properties (foci, directrices) to solve for unknown distances and positions.