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## High school geometry

### 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 parenthesis, x, minus, h, right parenthesis, squared, plus, left parenthesis, y, minus, k, right parenthesis, squared, equals, r, squared. When the equation is in this form, the center of the circle is located at left parenthesis, h, comma, k, right parenthesis and has a radius of r.
For example, the equation left parenthesis, x, plus, 2, right parenthesis, squared, plus, left parenthesis, y, minus, 3, right parenthesis, squared, equals, 16 represents a circle with center left parenthesis, minus, 2, comma, 3, right parenthesis 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 parenthesis, x, comma, y, right parenthesis on the parabola must be equidistant from the focus and directrix.
Consider, for example, the parabola whose focus is at left parenthesis, minus, 3, comma, 6, right parenthesis and directrix is y, equals, 4. We start by assuming a general point on the parabola left parenthesis, x, comma, y, right parenthesis.
Using the distance formula, we find that the distance between left parenthesis, x, comma, y, right parenthesis and the focus left parenthesis, minus, 3, comma, 6, right parenthesis is square root of, left parenthesis, x, plus, 3, right parenthesis, squared, plus, left parenthesis, y, minus, 6, right parenthesis, squared, end square root, and the distance between left parenthesis, x, comma, y, right parenthesis and the directrix y, equals, 4 is square root of, left parenthesis, y, minus, 4, right parenthesis, squared, end square root. On the parabola, these distances are equal:
\begin{aligned} \sqrt{(y-4)^2} &= \sqrt{(x+3)^2+(y-6)^2} \\\\ (y-4)^2 &= (x+3)^2+(y-6)^2 \\\\ \blueD{y^2}-8y\goldD{+16} &= (x+3)^2\blueD{+y^2}\maroonD{-12y}+36 \\\\ -8y\maroonD{+12y}&=(x+3)^2+36\goldD{-16} \\\\ 4y&=(x+3)^2+20 \\\\ y&=\dfrac{(x+3)^2}{4}+5\end{aligned}

## 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). •  Jimmy, totally agree. Won't help at all for putting cash in a register. The question is do you want to spend the next 40-50 years as a cashier? If so, cool but that may limit some of your choices in the future. Education provides opportunities in life. It can show how smart you are and how willing you are to work hard to achieve things. Honestly, there are things you will learn that you will never use but they are required stepping-stones to other opportunities. Think of it as an investment in your future. It's always better to be qualified for an opportunity that you decide to turn down than really want something but not have the required qualifications for it. When I was in college a professor said "How do you expect anyone to pay you if you don't know anything?" That really struck me as pretty solid advice. Some people will be rap stars or pro ballers or win the lottery but the vast, vast majority of us will have to work for a living. I can't stress enough to not limit yourself when you are young. Learn as much as you can and work hard at your education when you are young because changing your mind and trying to go back when you are older is a much, much harder approach if not impossible for some. I get it, it can suck to have to work hard at something that doesn't seem to have any practical application but just look at it as a challenge you have to accomplish to get to where you want to be.   