The circle is actually the inner part of the circumference, i.e., the enclosed area inside the circle frame.

Below are some examples of circles with different circumferences. The colored part in each represents the circle:

Question Types:

The circle is actually the inner part of the circumference, i.e., the enclosed area inside the circle frame.

Below are some examples of circles with different circumferences. The colored part in each represents the circle:

Question 1

M is the center of the circle.

Perhaps \( AB=CD \)

Question 2

There are only 4 radii in a circle.

Question 3

Where does a point need to be so that its distance from the center of the circle is the shortest?

Question 4

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

Question 5

Which figure shows the radius of a circle?

M is the center of the circle.

Perhaps $AB=CD$

CD is a diameter, since it passes through the center of the circle, meaning it is the longest segment in the circle.

AB does not pass through the center of the circle and is not a diameter, therefore it is necessarily shorter.

Therefore:

$AB\ne CD$

No

There are only 4 radii in a circle.

A radius is a straight line that connects the center of the circle with a point on the circle itself.

Therefore, the answer is incorrect, as there are infinite radii.

False

Where does a point need to be so that its distance from the center of the circle is the shortest?

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is less than the radius from the center of the circle will necessarily be inside the circle.

Inside

A point whose distance from the center of the circle is _______ than the radius, is outside the circle.

Let's remember that the circle is actually the inner part of the circumference, meaning the enclosed area within the frame of the circumference.

Therefore, a point whose distance is greater than the center of the circle will necessarily be outside the circle.

greater

Which figure shows the radius of a circle?

It is a straight line connecting the center of the circle to a point located on the circle itself.

Therefore, the diagram that fits the definition is c.

In diagram a, the line does not pass through the center, and in diagram b, it is a diameter.

Question 1

Which diagram shows a circle with a point marked in the circle and not on the circle?

Question 2

In which of the circles is the point marked in the circle and not on the circumference?

Question 3

Is it possible that the circumference of a circle is 8 meters and its diameter is 4 meters?

Question 4

_{A circle has the following equation:}_{\( x^2-8ax+y^2+10ay=-5a^2
\)}_{Point O is its center and is in the second quadrant (\( a\neq0 \))}

_{Use the completing the square method to find the center of the circle and its radius in terms of \( a \).}

Question 5

M is the center of the circle.

Perhaps \( MF=MC \)

Which diagram shows a circle with a point marked in the circle and not on the circle?

The interpretation of "in a circle" is inside the circle.

In diagrams a'-d' the point is on the circle, and in diagram c' the point is outside the circle.

In which of the circles is the point marked in the circle and not on the circumference?

Let's remember that the circular line draws the shape of the circle, and the inner part is called a disk.

Therefore, in diagram B, the point is located in the inner part, meaning inside the disk.

Is it possible that the circumference of a circle is 8 meters and its diameter is 4 meters?

To calculate, we will use the formula:

$\frac{P}{2r}=\pi$

Pi is the ratio between the circumference of the circle and the diameter of the circle.

The diameter is equal to 2 radii.

Let's substitute the given data into the formula:

$\frac{8}{4}=\pi$

$2\ne\pi$

Therefore, this situation is not possible.

Impossible

_{A circle has the following equation:}_{$x^2-8ax+y^2+10ay=-5a^2$}_{Point O is its center and is in the second quadrant ($a\neq0$)}

_{Use the completing the square method to find the center of the circle and its radius in terms of $a$.}

**Let's recall** that ** the equation of a circle with its center at **$O(x_o,y_o)$

$(x-x_o)^2+(y-y_o)^2=R^2$Now, ** let's now have a look at the equation for the given circle**:

$x^2-8ax+y^2+10ay=-5a^2$

We will try rearrange this equation to match** the circle equation**, or in other words we will ensure that on the left side is the sum of two squared binomial expressions, one for x and one for y.

We will do this using the "completing the square" method:

L**et's recall **the __short formula for squaring a binomial:__

$(c\pm d)^2=c^2\pm2cd+d^2$We'll deal ** separately **with the part of the equation related to x in the equation (underlined):

$\underline{ x^2-8ax}+y^2+10ay=-5a^2$

We'll isolate these two terms from the equation __and deal with them separately__.

We'll present these terms in a form **similar** to the form of the first two terms in the shortcut formula (we'll choose the **subtraction** form of the binomial squared formula since the term in the first power ** we are dealing with is**$8ax$, which has a negative sign):

$\underline{ x^2-8ax} \textcolor{blue}{\leftrightarrow} \underline{ c^2-2cd+d^2 }\\ \downarrow\\ \underline{\textcolor{red}{x}^2\stackrel{\downarrow}{-2 }\cdot \textcolor{red}{x}\cdot \textcolor{green}{4a}} \textcolor{blue}{\leftrightarrow} \underline{ \textcolor{red}{c}^2\stackrel{\downarrow}{-2 }\textcolor{red}{c}\textcolor{green}{d}\hspace{2pt}\boxed{+\textcolor{green}{d}^2}} \\$Notice that compared to the short formula (which is on the right side of the blue arrow in the previous calculation), we are actually making the comparison:

$\begin{cases} x\textcolor{blue}{\leftrightarrow}c\\ 4a\textcolor{blue}{\leftrightarrow}d
\end{cases}$ Therefore, if we want to get a squared binomial form from these two terms (underlined in the calculation), we will need to add the term$(4</span><span class="katex">a)^2$, __but we don't want to change the value of the expression, and therefore we will also subtract this term from the expression.__

That is, __we will add and subtract the term (or expression) we need to "complete" to the binomial squared form__,

In the following calculation, the "trick" is highlighted (two lines under the term we added and subtracted from the expression),

Next, **we'll put the expression in the squared binomial form** the appropriate expression (highlighted with colors) and in the last stage we'll simplify the expression:

$x^2-2\cdot x\cdot 4a\\ x^2-2\cdot
x\cdot4a\underline{\underline{+(4a)^2-(4a)^2}}\\ \textcolor{red}{x}^2-2\cdot \textcolor{red}{x}\cdot
\textcolor{green}{4a}+(\textcolor{green}{4a})^2-16a^2\\
\downarrow\\ \boxed{ (\textcolor{red}{x}-\textcolor{green}{4a})^2-16a^2}\\$**Let's summarize** the steps we've taken so far for the expression with x.

We'll do this **within the given equation**:

$x^2-8ax+y^2+10ay=-5a^2 \\ \textcolor{red}{x}^2-2\cdot \textcolor{red}{x}\cdot\textcolor{green}{4a}\underline{\underline{+\textcolor{green}{(4a)}^2-(4a)^2}}+y^2+10ay=-5a^2\\ \downarrow\\ (\textcolor{red}{x}-\textcolor{green}{4a})^2-16a^2+y^2+10ay=-5a^2\\$We'll continue and do the same thing for the expressions with y in the resulting equation:

(Now we'll choose **the addition form** of the squared binomial formula since the term in the first power ** we are dealing with **$10ay$ has a positive sign)

$(x-4a)^2-16a^2+\underline{y^2+10ay}=-5a^2\\ \downarrow\\ (x-4a)^2-16a^2+\underline{y^2+2\cdot y \cdot 5a}=-5a^2\\ (x-4a)^2-16a^2+\underline{y^2+2\cdot y \cdot 5a\underline{\underline{+(5a)^2-(5a)^2}}}=-5a^2\\ \downarrow\\ (x-4a)^2-16a^2+\underline{\textcolor{red}{y}^2+2\cdot\textcolor{red}{ y}\cdot \textcolor{green}{5a}+\textcolor{green}{(5a)}^2-25a^2}=-5a^2\\ \downarrow\\ (x-4a)^2-16a^2+(\textcolor{red}{y}+\textcolor{green}{5a})^2-25a^2=-5a^2\\ \boxed{(x-4a)^2+(y+5a)^2=36a^2}$In the last step, we move the free numbers to the second side and combine like terms.

Now that ** the given circle equation is in the form of the general circle equation** mentioned earlier, we can easily extract both the center of the given circle and its radius:

$(x-\textcolor{purple}{x_o})^2+(y-\textcolor{orange}{y_o})^2=\underline{\underline{R^2}} \\ \updownarrow \\ (x-\textcolor{purple}{4a})^2+(y+\textcolor{orange}{5a})^2=\underline{\underline{36a^2}}\\ \downarrow\\ (x-\textcolor{purple}{4a})^2+(y\stackrel{\downarrow}{- }(-\textcolor{orange}{5a}))^2=\underline{\underline{36a^2}}\\$

In the last step, we made sure to get the exact form of the general circle equation—that is, where only **subtraction** is performed within the squared expressions (emphasized with an arrow)

**Therefore, we can conclude that the center of the circle is at:**$\boxed{O(x_o,y_o)\leftrightarrow O(4a,-5a)}$ and extract the **radius of the circle** by solving a simple equation:

$R^2=36a^2\hspace{6pt}\text{/}\sqrt{\hspace{4pt}}\\ \rightarrow \boxed{R=\pm6a}$

Remember that the radius of the circle, by its definition ** is the distance** between any point on the diameter and the center of the circle. Since it is positive,

To do this, we will use the remaining information we haven't used yet—which is __that the center of the given circle O is in the second quadrant.__

That is:

O(x_o,y_o)\leftrightarrow x_o<0,\hspace{4pt}y_o>0 (Or in words: the x-value of the circle's center is negative and the y-value of the circle's center is positive)

Therefore, it must be true that:

\begin{cases} x_o<0\rightarrow (x_o=4a)\rightarrow 4a<0\rightarrow\boxed{a<0}\\ y_o>0\rightarrow (y_o=-5a)\rightarrow -5a>0\rightarrow\boxed{a<0} \end{cases}

We concluded that a<0 and since the radius of the circle is positive we conclude that necessarily:

$\rightarrow \boxed{R=-6a}$**Let's summarize:**

$\boxed{O(4a,-5a), \hspace{4pt}R=-6a}$__Therefore, the correct answer is answer d.__

_{$O(4a,-5a),\hspace{4pt}R=-6a$}

M is the center of the circle.

Perhaps $MF=MC$

Yes

Question 1

M is the center of the circle.

In the figure we observe 3 diameters?

Question 2

In which of the circles is the segment drawn the radius?

Question 3

Is there sufficient data to determine that

\( GH=AB \)

Question 4

In which of the circles is the center of the circle marked?

Question 5

Perhaps \( P=\pi\times EF \)

M is the center of the circle.

In the figure we observe 3 diameters?

No

In which of the circles is the segment drawn the radius?

Is there sufficient data to determine that

$GH=AB$

No

In which of the circles is the center of the circle marked?

Perhaps $P=\pi\times EF$

Yes