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### Course: High school geometry>Unit 7

Lesson 4: Focus and directrix of a parabola

# Conic sections FAQ

## What are conic sections?

Conic sections are the shapes you get when you slice a cone at different angles. There are four types of conic sections: ellipses, hyperbolas, parabolas, and circles.

## Where are conic sections used in the real world?

Conic sections show up in a lot of places! For example, the orbits of planets around the sun are elliptical. Hyperbolas are often used in the design of telescopes and antennas. Parabolas are important in physics, as they describe the shape of projectiles in flight.

## What is the standard equation of a circle?

The standard equation of a circle looks like $\left(x-h{\right)}^{2}+\left(y-k{\right)}^{2}={r}^{2}$. When the equation is in this form, the center of the circle is located at $\left(h,k\right)$ and has a radius of $r$.
For example, the equation $\left(x+2{\right)}^{2}+\left(y-3{\right)}^{2}=16$ represents a circle with center $\left(-2,3\right)$ and radius $4$.

## What do we mean by the focus and directrix in parabolas?

A parabola can be graphed as the set of all points that are equidistant from a fixed point (the "focus") and a fixed line (the "directrix").

## How can we find the equation of a parabola from its focus and directrix?

We use the fact that any point $\left(x,y\right)$ on the parabola must be equidistant from the focus and directrix.
Consider, for example, the parabola whose focus is at $\left(-3,6\right)$ and directrix is $y=4$. We start by assuming a general point on the parabola $\left(x,y\right)$.
Using the distance formula, we find that the distance between $\left(x,y\right)$ and the focus $\left(-3,6\right)$ is $\sqrt{\left(x+3{\right)}^{2}+\left(y-6{\right)}^{2}}$, and the distance between $\left(x,y\right)$ and the directrix $y=4$ is $\sqrt{\left(y-4{\right)}^{2}}$. On the parabola, these distances are equal:
$\begin{array}{rl}\sqrt{\left(y-4{\right)}^{2}}& =\sqrt{\left(x+3{\right)}^{2}+\left(y-6{\right)}^{2}}\\ \\ \left(y-4{\right)}^{2}& =\left(x+3{\right)}^{2}+\left(y-6{\right)}^{2}\\ \\ {y}^{2}-8y+16& =\left(x+3{\right)}^{2}+{y}^{2}-12y+36\\ \\ -8y+12y& =\left(x+3{\right)}^{2}+36-16\\ \\ 4y& =\left(x+3{\right)}^{2}+20\\ \\ y& =\frac{\left(x+3{\right)}^{2}}{4}+5\end{array}$

## Want to join the conversation?

• How will conic sections help me in the real world? I mean, facts like "The orbits of planets around the sun are elliptical. Hyperbolas are often used in the design of telescopes and antennas." are factinating, but it's not like I will use conic sections when I get a job. I definety don't need conic sections when I add money on a cash register. Someone please answer my question(Can't believe I'm the first one asking a question on this article).