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### Course: Algebra (all content)>Unit 17

Lesson 11: Identifying conic sections from their expanded equations

# Conic section from expanded equation: hyperbola

Sal manipulates the equation 4y^2-50x=25x^2+16y+109 in order to find that it represents a hyperbola. Created by Sal Khan.

## Want to join the conversation?

• Sal, In my upper level math class we are deriving the standard formulas for all the conic sections from the distance formula. I am stuck on the hyperbola. I can derive the equations for the asymptotes but I'm having trouble deriving the actual equation for the hyperbola. Can you help? Is there a video I can watch to give me some hint? thanks
• what are identity, contradiction and conditional equations?
AN EQUATION THAT HAS NO ANSWERS (NULL SET or NO SOLUTION
Example;
3(x+2)=3x-5
3x+6=3x-5
Subtract (3x) from both sides
6=-5
False Statement
No Solution

Conditional Equation:
AN EQUATION THAT IS TRUE BASED "ON THE CONDITION" THAT THE SOLUTION IS X=(A CERTAIN) THAT MAKES THE EQUATION TRUE
Example;
3(x+2)=2x-5
3x+6=2x-5
Subtract 3x from both sides AND Subtract 6 from both sides
x=-11
The equation is true based on the condition that x=-11

Identity:
AN EQUATION THAT HAS ALL REAL NUMBERS AS ITS ANSWERS. At some step in the solving of the equation you will get the same IDENTICAL terms on both sides of the equation.
Example;
3(x+2)=3x+6
3x+6=3x+6(Note the same terms on both sides)
COULD STOP HERE and say the solution is ALL REAL NUMBERS
If you Subtract 3x from both sides AND Subtract 6 from both sides you will get
0=0
A TRUE Statement
Solution ::::All Real Numbers

Reference:
• How do you know for sure if it's a conic? I mean, what if you try to simplify it and then you can't multiply it by a number so that it has the form ((x+a)^2)/b)+((y+c)^2)/d)=1? For example, what if, at , the number on the right wasn't 100? What would you do then?
• It doesnt matter if it is some other number than 100. It can even be negative but not zero thought.
• Did Sal define what an asymptote was in an earlier video?
(1 vote)
• It's the video:"Conic Sections: Intro to Hyperbolas : Introduction to the hyperbola" beginning about

He doesn't explain them in details, but once the conic sections videos starts he mentions a couple of times, that those are important when dealing with hyperbolas. They are the lines to which the hyperbola gets closer and closer without ever touching it (see first and second video on conic sections)
• So, at , Sal mentions that we can determine what kind of conic the equation will be by the x^2 and y^2 coefficients. WHY is it a hyperbolic equation if one coefficient is negative and the other is positive?
• it is just a characteristic of hyperbolic equations. This differentiates hyperbola from the ellipse's equation( having both x^2 and y^2 positive)
• Are the two asymptotes always intersecting?
• Yes, they always intersect at the center of the hyperbola, which is also the midpoint of the two focuses or the two vertexes.
• what do I do when there is no y^2?
(1 vote)
• If there is no y^2 term at all then it is a parabola. The same is true if there is no x^2 term.
• How do you tell if the equation is a hyperbola, parabola, ellipse, or circle? I understand how to solve them if I know which of the four it's asking for, but not when I need to differentiate it and solve it on my own.
• Ellipse is (x+h)^2 / a^2 + (y+v)^2 / b^2 = 1 where a and b determine the horizontal and vertical radii respectively, keeping in mind a is int he denominator under x and b is under y. h moves the ellipse left and right, where a negative h moves it to the right, and a positive h moves it left. Similarly a positive v moves it down and a negative v moves it up. Specifically h and v move the center of the ellipse, or more accurately where the radii are measured from. It's usually best to find this "center" point first. A shortcut is that the point is (-h, -v).

For this equation and all others, if it is not equal to 1 you can make it equal to 1, it just changes a and b. You need to have it equal to 1 though in this standard form. Say it is equal to c which is not 1 or 0, then you do this.

(x+h)^2 / a^2 + (y+v)^2 / b^2 = c divide both sides by c

(x+h)^2 / ca^2 + (y+v)^2 / cb^2 = 1

So now the two radii are not a and b, but sqrt(c)a and sqrt(c)b

A circle is a special version of an ellipse, basically where a = b. let's call this c. (x+h)^2 / c^2 + (y+v)^2 / c^2 = 1. In this form you can change it to the more normal form of a circle.

(x+h)^2 / c^2 + (y+v)^2 / c^2 = 1
1/c^2 [(x+h)^2 + (y+v)^2] = 1
(x+h)^2 + (y+v)^2 = c^2

So c is the radius all around.

A hyperbola is also similar in equation to an ellipse, there is just a minus sign in the equation. Just one minus though. So that means a hyperbola is either (x+h)^2 / a^2 - (y+v)^2 / b^2 = 1 or (y+v)^2 / b^2 - (x+h)^2 / a^2 = 1. If it's -(y+v)^2 / b^2 - (x+h)^2 / a^2 = 1 with two minuses on the left or (y+v)^2 / b^2 + (x+h)^2 / a^2 = -1 where there is no minus on the left but equal to -1 on the right there is no solution.

A parabola meanwhile is an equation where only one variable is squared. so you could put it into the form x = ay^2 + by or y = cx^2 + dx. keep in mind though b and d can be 0, and there are no powers above 2.

• At shouldn't it be y=3 and y=-7?