Derivation of Hyperbolae

Let the coordinates of a point on a hyperbola be (x,y)(x,y), the distance from the center of the hyperbola to either vertex aa and the distance from the center of the hyperbola to either focus cc.

The difference of the distances from a point on the hyperbola to the two foci is equivalent to both (x+c)2+y2(xc)2+y2\sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2} and 2a2a, thus (x+c)2+y2(xc)2+y2=2a\sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2}=2a. The equation is true for all points (x,y)(x,y) on the hyperbola because the difference is invariant of (x,y)(x,y).

Derivation:

(x+c)2+y2(xc)2+y2=2a(x+c)2+y2=2a+(xc)2+y2(x+c)2+y2=4a2+4a(xc)2+y2+(xc)2+y2x2+2cx+c2+y2=4a2+4a(xc)2+y2+x22cx+c2+y22cx=4a2+4a(xc)2+y22cx4cx4a2=4a(xc)2+y2cxa2=a(xc)2+y2(cxa2)2=a2(x22cx+c2+y2)c2x22a2cx+a4=a2x22a2cx+a2c2+a2y2c2x2+a4=a2x2+a2c2+a2y2c2x2a2x2a2y2=a2c2a4x2(c2a2)a2y2=a2(c2a2)Let a2+b2=c2b2=c2a2x2b2a2y2=a2b2\begin{aligned} \sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2}&=2a\\ \sqrt{(x+c)^2+y^2}&=2a+\sqrt{(x-c)^2+y^2}\\ (x+c)^2+y^2&=4a^2+4a\sqrt{(x-c)^2+y^2}+(x-c)^2+y^2\\ x^2+2cx+c^2+y^2&=4a^2+4a\sqrt{(x-c)^2+y^2}+x^2-2cx+c^2+y^2\\ 2cx&=4a^2+4a\sqrt{(x-c)^2+y^2}-2cx\\ 4cx-4a^2&=4a\sqrt{(x-c)^2+y^2}\\ cx-a^2&=a\sqrt{(x-c)^2+y^2}\\ \left(cx-a^2\right)^2&=a^2\left(x^2-2cx+c^2+y^2\right)\\ c^2x^2-2a^2cx+a^4&=a^2x^2-2a^2cx+a^2c^2+a^2y^2\\ c^2x^2+a^4&=a^2x^2+a^2c^2+a^2y^2\\ c^2x^2-a^2x^2-a^2y^2&=a^2c^2-a^4\\ x^2\left(c^2-a^2\right)-a^2y^2&=a^2\left(c^2-a^2\right)\quad\text{Let }a^2+b^2=c^2\Rightarrow b^2=c^2-a^2\\ x^2b^2-a^2y^2&=a^2b^2\\ \end{aligned}

x2a2y2b2=1\therefore\boxed{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}

Note by Gordon Chan
1 week, 2 days ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...