Circle with center at the origin

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# Circle with center at the origin
## Analytic Geometry
### Arturo Sánchez González
### <a href="mailto:arturo.sanchez@upaep.mx" class="email">arturo.sanchez@upaep.mx</a>
### August 2021

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# Again... the circle

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# Geometric definition of the circle

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</br>
</br>

<span style="font-size:30px">A <span style="color:green">**CIRCLE**</span> is

<span style="font-size:33px">_**the set of all points in the plane** that are <span style="color:blue">a fixed distance</span> (the <span style="color:blue">radius</span>) from <span style="color:red">a fixed point</span> (the <span style="color:red">center</span>)_
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</br>

<img src="circle.png" width="110%" style="display: block; margin: auto;" />
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# How do we transform the previous definition into an equation?

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<span style="color:white;font-size:35px"> First, let's suppose that we have a **circle with center in the origin `\(O(0,0)\)` and radius** `\(r\)`

</br>
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- <span style="color:white;font-size:35px"> Let's take a point `\(A(x,y)\)` in the circle.

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- <span style="color:white;font-size:35px"> Then, we know that
  $$
  d(A,O) = r
  $$
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- <span style="color:white;font-size:35px"> This implies that
  $$
    \sqrt{(x-0)^2 + (y-0)^2} = r
  $$

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- <span style="color:white;font-size:35px"> If we make the necessary reductions:
 $$
 \sqrt{x^2 + y^2}=r
 $$

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- <span style="color:white;font-size:35px"> Finally, if we take squares in both sides, we get</span>
 <span style="font-size:35px"> $$
  \color{yellow}{x^2 + y^2 = r^2}
 $$
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<span style="color:white;font-size:30px">This is called the 
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<span style="color:white;font-size:35px"><b>CANONICAL EQUATION OF A CIRCLE WITH RADIUS `\(r\)` CENTERED AT THE ORIGIN
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# Problem 1 (a)
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<span style="color:white;font-size:35px">Find the equation of a circle of radius `\(5\)` and center at the origin.

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# Big question:

<span style="font-size:40px"> **Why is important to have an equation that represents a figure ?**

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# Main reasons to use equations

</br>

- <span style="font-size:30px">We have a compact representation of all the points in the plane that lie on the figure.

</br>
  - <span style="font-size:30px">The equation allows us to confirm if a point lies on the figure just by making a substitution.

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# Problem 1 (b)

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<span style="color:white;font-size:35px">Does the point `\((-3, -4)\)` lie on the circle of radius `\(5\)` and center in the origin?

<span style="color:white;font-size:35px"> What about the point `\((1,-5)\)`?

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# Problem 1 (c)

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<span style="color:white;font-size:35px">Find the equation of a circle of radius `\(\sqrt{23}\)` and center at the origin.

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# Problem 1 (d)
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<span style="color:white;font-size:35px">Find the elements of the circle with equation `\(x^2 + y^2 - 64 =0\)`.

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# Problem 2

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<span style="color:white;font-size:35px"> Given the equation `\(x^2 + y^2 = 23\)`, determine the elements of the circle. Draw this circle.

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# Problem 3

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<span style="color:white;font-size:35px"> Find the equation of the circle centered at the origin that passes through `\((5,8)\)`.

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# Problem 4
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<span style="color:white;font-size:35px"> Determine the figure represented by the equation `\(6x^2+6y^2=150\)`. Make a graphical representation of this figure.

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# Homework

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<span style="color:white;font-size:30px"> **Problem A**

<span style="color:white;font-size:35px"> Let's consider a circle with center at the origin and radius `\(13\)`. Determine its canonical equation. Draw this circle.

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<span style="color:white;font-size:30px"> **Problem B**

<span style="color:white;font-size:35px"> You draw a circle that is centered at the origin. You measure the diameter of the circle to be `\(32\)` units. Does the point `\((14,8)\)` lie on the circle? Draw this circle.