1- معادله دایره‌ای را بنویسید که:

الف) $O(1,1)$ مرکز آن و $A(3,2)$ نقطه‌ای از آن باشد.

$r = OA = \sqrt {{2^2} + {1^2}}  = \sqrt 5$

${(x - 1)^2} + {(y - 1)^2} = 5$

ب) $O(2,1)$ مرکز آن بوده و بر خط $3x + 4y = 0$ مماس باشد.

$r = \frac{{\left| {3(2) + 4(1) + 0} \right|}}{{\sqrt {9 + 16} }} = \frac{{10}}{5} = 2$

${(x - 2)^2} + {(y - 1)^2} = 4$

پ) $O( - 1, - 1)$ مرکز آن بوده و روی خط $x + y = 1$ وتری به طول 2 ایجاد کند.

$OH = \frac{{\left| { - 1 - 1 - 1} \right|}}{{\sqrt {1 + 1} }} = \frac{3}{{\sqrt 2 }} = \frac{{3\sqrt 2 }}{2}$

${r^2} = {1^2} + {\left( {\frac{{3\sqrt 2 }}{2}} \right)^2} = 1 + \frac{9}{2} = \frac{{11}}{2}$

${(x + 1)^2} + {(y + 1)^2} = \frac{{11}}{2}$

ت) خطوط $x + y = 1$ و $x - y = 3$ شامل قطرهایی از آن بوده و خط $4x + 3y = 6$ بر آن مماس باشد.

$\left\{ \begin{gathered}   x + y = 1 \hfill \\   x - y = 3 \hfill \\  \end{gathered}  \right. \Rightarrow 2x = 4 \to x = 2,y =  - 1$

$o(2, - 1)$

$r = \frac{{\left| {4(2) + 3( - 1) - 6} \right|}}{{\sqrt {16 + 9} }} = \frac{1}{5}$

${(x - 2)^2} + {(y + 1)^2} = \frac{1}{{25}}$

ث) از نقاط $A(1,2)$ و $B(3,0)$ بگذرد و $y = 2x - 1$ شامل قطری از آن باشد.

$o\left| \begin{gathered}   \,\,\,\,\alpha  \hfill \\   2\alpha  - 1 \hfill \\  \end{gathered}  \right.$

$OA = OB = r = \sqrt {{{(\alpha  - 1)}^2} + {{(2\alpha  - 1 - 2)}^2}}  = \sqrt {{{(\alpha  - 3)}^2} + {{(2\alpha  - 1 - 0)}^2}}$

$\cancel{{{\alpha ^2}}} - 2\alpha  + 1 + \cancel{{4{\alpha ^2}}} - 12\alpha  + 9 = \cancel{{{\alpha ^2}}} - 6\alpha  + 9 + \cancel{{4{\alpha ^2}}} - 4\alpha  + 1$

$- 14\alpha  + \cancel{{10}} =  - 10\alpha  + \cancel{{10}} \to \alpha  = 0$

$o(0, - 1) \Rightarrow OA = OB = r = \sqrt {10}$

${X^2} + {(y + 1)^2} = 10$

2- حدود $a$ را طوری به‌دست آورید که ${x^2} + {y^2} - 3x + 5y + a = 0$ بتواند معادلهٔ یک دایره باشد.

3- وضعیت هر یک از نقاط $A( - 1, - 1)$ و $B(1, - 2)$ و $C(2,3)$ و $D(4, - 1)$ را نسبت به دایره ${x^2} + {y^2} - 2x + 4y - 5 = 0$ تعیین کنید.

${X^2} + {y^2} - 2x + 4y - 5 = 0 \to O\left| \begin{gathered}   1 \hfill \\    - 2 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,\,\,\,r = \frac{{\sqrt {4 + 16 + 20} }}{2} = \sqrt {10}$

نقطه درون دایره است $OA = \sqrt {4 + 1}  = \sqrt 5 \, < r \to$

نقطه روی مرکز دایره است $OB = \sqrt {0 + 0}  = 0\, < r \to$

نقطه روی محیط دایره است $OD = \sqrt {9 + 1}  = \sqrt {10}  = r \to$

4- وضعیت هر یک از جفت دایره‌های زیر را نسبت به هم مشخص کنید:

${x^2} + {y^2} = 4\,\,\,\,\,,\,\,\,\,\,{x^2} + {y^2} - 2x = 4$ (الف

${x^2} + {y^2} = 4\,\,\,\,\,\,\,\,O\left| \begin{gathered}   0 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,r = 2$

${x^2} + {y^2} - 2x = 4\,\,\,\,\,\,\,\,O'\left| \begin{gathered}    - 1 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,r' = \frac{1}{2}\sqrt {{{( - 2)}^2} + {0^2} - 4( - 4)}  = \frac{1}{2} \times 2\sqrt 5  = \sqrt 5$

دو دایره متقاطع هستند $OO' =  \to \left| {2 - \sqrt 5 } \right| \lt 1 \lt 2 + \sqrt 5  \to \left| {r - r'} \right| \lt OO' \lt r + r' \to$

${x^2} + {(y - 1)^2} = 1\,\,\,\,\,,\,\,\,\,\,{(x - 1)^2} + {y^2} = 1$ (ب

${x^2} + {(y - 1)^2} = 1\,\,\,\,\,\,\,O\left| \begin{gathered}   0 \hfill \\   1 \hfill \\  \end{gathered}  \right.\,\,\,\,r = 1$

${(x - 1)^2} + {y^2} = 1\,\,\,\,\,\,O'\left| \begin{gathered}   1 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,\,r' = 1$

دو دایره متقاطع هستند  $OO' = \sqrt 2  \to \left| {1 - 1} \right| \lt \sqrt 2  \lt 1 + 1 \to \left| {r - r'} \right| \lt OO' \lt r + r' \to$

${x^2} + {y^2} = 1\,\,\,\,\,,\,\,\,\,\,{x^2} + {y^2} - 3\sqrt 2 x - 3\sqrt 2 y + 5 = 0$ (ج

${x^2} + {y^2} = 1\,\,\,\,\,\,O\left| \begin{gathered}   0 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,\,r = 1$

${x^2} + {y^2} - 3\sqrt 2 x - 3\sqrt 2 y + 5 = 0\,\,\,\,\,\,O'\left| \begin{gathered}   1 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,\,r' = \frac{1}{2}\sqrt {18 + 18 - 20}  = 2$

دو دایره مماس هستند $OO' = \sqrt {\frac{{18}}{4} + \frac{{18}}{4}}  = \sqrt 9  = 3 \to 3 = 1 + 2 \to OO' = r + r' \to$

${x^2} + {y^2} = 1\,\,\,\,\,,\,\,\,\,\,{x^2} + {y^2} - 6x - 2y + 9 = 0$ (د

${x^2} + {y^2} = 1\,\,\,\,\,\,\,\,\,O\left| \begin{gathered}   0 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,r = 1$

${x^2} + {y^2} - 6x - 2y + 9 = 0\,\,\,\,\,\,\,\,\,O'\left| \begin{gathered}   3 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,r' = \frac{1}{2}\sqrt {36 + 4 - 36}  = 1$

دو دایره متخارج هستند  $OO' = \sqrt {9 + 1}  = \sqrt {10}  \to \sqrt {10}  \gt 1 + 1 \to OO' \gt r + r' \to$

5- نقاط $A( - 1, - 1)$ و $B(1,1)$ و $C(1, - 3)$ رئوس مثلث $ABC$ هستند. معادلهٔ دایره محیطی مثلث $ABC$ را بنویسید. سپس معادلهٔ مماس بر این دایره را در رأس $B$ به‌دست آورید.

${x^2} + {y^2} + ax + by + c = 0$

$\left\{ \begin{gathered}   1 + 1 - a - b + c - 0\,\,\,\,\, \hfill \\   1 + 1 + a + b + c = 0\,\,\,\, \hfill \\   1 + 9 + a - 3b + c = 0 \hfill \\  \end{gathered}  \right. \to \left\{ \begin{gathered}    - a - b + c =  - 2\,\,\,(1) \hfill \\   a + b + c =  - 2\,\,\,\,\,\,\,(2) \hfill \\   a - 3b + c =  - 1\,\,\,\,\,\,(3) \hfill \\  \end{gathered}  \right.$

$\xrightarrow{{(1) + (2)}}c =  - 2\xrightarrow{{(2),(3)}}\left\{ \begin{gathered}   a + b - 2 =  - 2\,\,\,\,\, \hfill \\   a - 3b - 2 =  - 10 \hfill \\  \end{gathered}  \right. \Rightarrow a =  - 2,b = 2$

$\boxed{{x^2} + {y^2} - 2x + 2y - 2 = 0}\,\,\,\,\,O\left| \begin{gathered}   \,\,1 \hfill \\    - 1 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,r = \frac{1}{2}\sqrt {4 + 4 + 8}  = 2$

$\to m - 0$ تعریف نشده $O\left| \begin{gathered}   \,\,1 \hfill \\    - 1 \hfill \\  \end{gathered}  \right.\,\,\,\,B\left| \begin{gathered}   1 \hfill \\   1 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,{m_{OB}} =$

مماس در $B$       $\xrightarrow[{y = {y_B}}]{{m = 0}}y = 1$

6- وضعیت هر یک از خطوط و دایره‌های زیر را نسبت به هم مشخص کنید:

$3x + 4y = 0\,\,\,\,\,,\,\,\,\,\,{x^2} + {y^2} - 4x - 4y + 7 = 0$ (الف

$\left\{ \begin{gathered}   3x + 4y = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\   {x^2} + {y^2} - 4x + 4y + 7 = 0 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,\,\,\,\,\,O\left| \begin{gathered}   2 \hfill \\   2 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,r = \frac{1}{2}\sqrt {16 + 16 - 28}  = 1$

غیر متقاطع        $d = \frac{{\left| {3(2) + 4(2)} \right|}}{{\sqrt {9 + 16} }} = \frac{{14}}{5} \gt 1\,\,\,\,\,\,\,\,\,\,\,\,d \gt r$

$x + y = 2\,\,\,\,\,,\,\,\,\,\,{x^2} + {y^2} = 2$ (ب

$\left\{ \begin{gathered}   x + y = 2\,\,\,\, \hfill \\   {x^2} + {y^2} = 2 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,\,\,\,\,\,O\left| \begin{gathered}   0 \hfill \\   0 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,r = \sqrt 2$

خط بر دایره مماس است       $d = \frac{{\left| {1(0) + 1(0) - 2} \right|}}{{\sqrt {1 + 1} }} = \sqrt 2 \,\,\,\,\,\,\,\,\,d = r$

$x + y = 1\,\,\,\,\,,\,\,\,\,\,{x^2} + {y^2} - 2x - 2y = 2$ (ج

$\left\{ \begin{gathered}   x + y = 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\   {x^2} + {y^2} - 2x - 2y = 2 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,\,\,\,\,\,O\left| \begin{gathered}   1 \hfill \\   1 \hfill \\  \end{gathered}  \right.\,\,\,\,\,\,\,\,r = \frac{1}{2}\sqrt {4 + 4 + 8}  = 2$

خط و دایره متقاطع‌اند.        $d = \frac{{\left| {1(1) + 1(1) - 1} \right|}}{{\sqrt {1 + 1} }} = \frac{{\sqrt 2 }}{2}\,\,\,\,\,\,\,\,\,d \lt r$