Memahami Persamaan Lingkaran dan Unsur-unsurnya

Table of Contents

Konsep Lingkaran

1. Konsep Lingkaran dalam Roda Mobil

Roda mobil adalah contoh nyata penerapan lingkaran dalam kehidupan sehari-hari. Unsur-unsur lingkaran seperti jari-jari (r), diameter (d), titik pusat, dan keliling dapat diamati langsung pada roda.

• Titik Pusat (Center): Pusat roda.

• Jari-jari (r): Jarak dari pusat roda ke tepi ban.

• Diameter (d): Jarak lurus melintasi roda (d = 2r).

2. Persamaan Lingkaran

a. Jika pusatnya (0,0) dan jari-jari itu r

bentuk persamaannya x2 + y2 = r2.

b. Jika pusatnya (a,b) dan jari-jari itu r,

Persamaan umum lingkaran dengan pusat di titik $(a,b)$ dan jari-jari $r$ adalah:

$$<span\; style="font-family:\; arial;">(</span>\mathrm{<span\; style="font-family:\; arial;">x</span>}<span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">a</span>}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span><span\; style="font-family:\; arial;">(</span>\mathrm{<span\; style="font-family:\; arial;">y</span>}<span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">b</span>}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">=</span>{\mathrm{<span\; style="font-family:\; arial;">r</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}$$

Contoh Penerapan:
Jika pusat roda berada di titik $(0,0)$ (misalnya saat mobil diam), maka persamaan lingkarannya:

$${\mathrm{<span\; style="font-family:\; arial;">x</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">y</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">=</span>{\mathrm{<span\; style="font-family:\; arial;">r</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}$$

3. Unsur-Unsur Lingkaran pada Roda

• Keliling (K): Panjang lintasan satu putaran roda.

  $$\mathrm{<span\; style="font-family:\; arial;">K</span>}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">2</span>}\mathrm{<span\; style="font-family:\; arial;">\pi </span>}\mathrm{<span\; style="font-family:\; arial;">r</span>}\text{<span style="font-family: arial;">atau</span>}\mathrm{<span\; style="font-family:\; arial;">K</span>}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">\pi </span>}\mathrm{<span\; style="font-family:\; arial;">d</span>}$$

  Contoh: Jika jari-jari roda $30\text{ cm}$, maka kelilingnya $60\pi \text{ cm}$.

• Luas (L): Area permukaan roda.

  $$\mathrm{<span\; style="font-family:\; arial;">L</span>}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">\pi </span>}{\mathrm{<span\; style="font-family:\; arial;">r</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}$$

• Kecepatan Mobil: Berhubungan dengan putaran roda per satuan waktu.

  $$\text{<span style="font-family: arial;">Kecepatan</span>}<span\; style="font-family:\; arial;">=</span>\text{<span style="font-family: arial;">Keliling</span>}<span\; style="font-family:\; arial;">\times </span>\text{<span style="font-family: arial;">Jumlah Putaran per Detik</span>}$$

 Contoh Soal

Sebuah roda mobil memiliki jari-jari $28\text{ cm}$. Tentukan:

1. Keliling roda:

   $$\mathrm{<span\; style="font-family:\; arial;">K</span>}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">2</span>}\mathrm{<span\; style="font-family:\; arial;">\pi </span>}<span\; style="font-family:\; arial;">\times </span>\mathrm{<span\; style="font-family:\; arial;">28</span>}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">56</span>}\mathrm{<span\; style="font-family:\; arial;">\pi </span>}\text{<span style="font-family: arial;"> cm</span>}$$

2. Persamaan lingkaran jika pusatnya di $(0,0)$:

   $${\mathrm{<span\; style="font-family:\; arial;">x</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">y</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">=</span>{\mathrm{<span\; style="font-family:\; arial;">28</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}\text{<span style="font-family: arial;">atau</span>}{\mathrm{<span\; style="font-family:\; arial;">x</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">y</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">784</span>}$$

4. Memahami Cara Mencari Titik Pusat dan Jari-Jari dari Persamaan Lingkaran

1. Bentuk Standar

   $$<span\; style="font-family:\; arial;">(</span>\mathrm{<span\; style="font-family:\; arial;">x</span>}<span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">a</span>}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span><span\; style="font-family:\; arial;">(</span>\mathrm{<span\; style="font-family:\; arial;">y</span>}<span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">b</span>}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">=</span>{\mathrm{<span\; style="font-family:\; arial;">r</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}$$

   • Titik pusat lingkaran: $(a,b)$

   • Jari-jari (r): $\sqrt{r^2}$ atau langsung $r$ jika persamaan sudah sederhana.

   Contoh:
   $(x-2{)}^2+(y+3{)}^2=16$

   • Pusat: $(2,-3)$

   • Jari-jari: $\sqrt{16}=4$

2. Bentuk Umum

   $${\mathrm{<span\; style="font-family:\; arial;">x</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">y</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">A</span>}\mathrm{<span\; style="font-family:\; arial;">x</span>}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">B</span>}\mathrm{<span\; style="font-family:\; arial;">y</span>}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">C</span>}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">0</span>}$$

   Untuk menemukan pusat dan jari-jari, lakukan melengkapkan kuadrat:

   • Langkah 1: Kelompokkan $x$ dan $y$:

     $${\mathrm{<span\; style="font-family:\; arial;">x</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">A</span>}\mathrm{<span\; style="font-family:\; arial;">x</span>}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">y</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">B</span>}\mathrm{<span\; style="font-family:\; arial;">y</span>}<span\; style="font-family:\; arial;">=</span><span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">C</span>}$$

   • Langkah 2: Lengkapi kuadrat untuk $x$ dan $y$:

     $$<span\; style="font-family:\; arial;">(</span>{\mathrm{<span\; style="font-family:\; arial;">x</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">A</span>}\mathrm{<span\; style="font-family:\; arial;">x</span>}<span\; style="font-family:\; arial;">+</span>{<span\; style="font-family:\; arial;">(</span>\frac{\mathrm{<span\; style="font-family:\; arial;">A</span>}}{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">)</span><span\; style="font-family:\; arial;">+</span><span\; style="font-family:\; arial;">(</span>{\mathrm{<span\; style="font-family:\; arial;">y</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">B</span>}\mathrm{<span\; style="font-family:\; arial;">y</span>}<span\; style="font-family:\; arial;">+</span>{<span\; style="font-family:\; arial;">(</span>\frac{\mathrm{<span\; style="font-family:\; arial;">B</span>}}{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">)</span><span\; style="font-family:\; arial;">=</span><span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">C</span>}<span\; style="font-family:\; arial;">+</span>{<span\; style="font-family:\; arial;">(</span>\frac{\mathrm{<span\; style="font-family:\; arial;">A</span>}}{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{<span\; style="font-family:\; arial;">(</span>\frac{\mathrm{<span\; style="font-family:\; arial;">B</span>}}{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}$$

   • Langkah 3: Ubah ke bentuk standar:

     $$<span\; style="font-family:\; arial;">(</span>\mathrm{<span\; style="font-family:\; arial;">x</span>}<span\; style="font-family:\; arial;">+</span>\frac{\mathrm{<span\; style="font-family:\; arial;">A</span>}}{\mathrm{<span\; style="font-family:\; arial;">2</span>}}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span><span\; style="font-family:\; arial;">(</span>\mathrm{<span\; style="font-family:\; arial;">y</span>}<span\; style="font-family:\; arial;">+</span>\frac{\mathrm{<span\; style="font-family:\; arial;">B</span>}}{\mathrm{<span\; style="font-family:\; arial;">2</span>}}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">=</span>\frac{{\mathrm{<span\; style="font-family:\; arial;">A</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">B</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">4</span>}\mathrm{<span\; style="font-family:\; arial;">C</span>}}{\mathrm{<span\; style="font-family:\; arial;">4</span>}}$$

   • Titik pusat: $(-\frac{A}{2},-\frac{B}{2})$

   • Jari-jari: $r=\sqrt{\frac{A^2+B^2-4C}{4}}$ atau $\frac{\sqrt{A^2+B^2-4C}}{2}$.

   Contoh:
   $x^2+y^2-4x+6y-3=0$

• $A=-4$, $B=6$, $C=-3$

• Pusat: $(-\frac{-4}{2},-\frac{6}{2})=(2,-3)$

• Jari-jari:

  $$\mathrm{<span\; style="font-family:\; arial;">r</span>}<span\; style="font-family:\; arial;">=</span>\sqrt{\frac{<span\; style="font-family:\; arial;">(</span><span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">4</span>}{<span\; style="font-family:\; arial;">)</span>}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">+</span>{\mathrm{<span\; style="font-family:\; arial;">6</span>}}^{\mathrm{<span\; style="font-family:\; arial;">2</span>}}<span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">4</span>}<span\; style="font-family:\; arial;">(</span><span\; style="font-family:\; arial;">-</span>\mathrm{<span\; style="font-family:\; arial;">3</span>}<span\; style="font-family:\; arial;">)</span>}{\mathrm{<span\; style="font-family:\; arial;">4</span>}}}<span\; style="font-family:\; arial;">=</span>\sqrt{\frac{\mathrm{<span\; style="font-family:\; arial;">16</span>}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">36</span>}<span\; style="font-family:\; arial;">+</span>\mathrm{<span\; style="font-family:\; arial;">12</span>}}{\mathrm{<span\; style="font-family:\; arial;">4</span>}}}<span\; style="font-family:\; arial;">=</span>\sqrt{\mathrm{<span\; style="font-family:\; arial;">16</span>}}<span\; style="font-family:\; arial;">=</span>\mathrm{<span\; style="font-family:\; arial;">4</span>}$$

Sebuah lingkaran memiliki titik pusat (2, 3) dan berdiameter 8 cm. Persamaan lingkaran tersebut adalah…

Karena d = 8, berarti r = 8/2 = 4, sehingga persamaan lingkaran yang terbentuk adalah:

(x – 2)² + (y – 3)² = 4²

x² – 4x + 4 + y² -6y + 9 = 16

x² + y² – 4x – 6y – 3 = 0

• $<span\; style="font-family:\; arial;"><br></span><br>$

Untuk selanjutnya coba dikerjakan soal dibawah ini