SAT-80: Completing the Square to Find a Circle's Center and Radius

SAT-80: Completing the Square to Find a Circle's Center and Radius

Description: Sometimes a circle is given expanded, like x² + y² + 6x − 4y − 3 = 0, hiding its center and radius. Completing the square for x and for y rewrites it into the standard form of SAT-79 so you can read them off.

Completing the square — the move

For a group like x² + bx, take half of b, square it, and add it to make a perfect square: x² + bx + (b/2)² = (x + b/2)². Whatever you add on one side you must balance on the other. (Oʻzbekcha: x² + bx ni toʻliq kvadratga keltirish uchun (b/2)² ni qoʻshamiz.)

The plan for a circle

1. Group the x-terms together and the y-terms together; move the constant to the right side.
2. Complete the square for x, and separately for y (add the same amounts to the right side).
3. Write each group as a square; read center (h, k) and radius √(right side).

(Oʻzbekcha: x va y hadlarini alohida guruhlab, har birini toʻliq kvadratga keltiramiz.)

Worked Example 1 — full process

Find the center and radius of x² + y² + 6x − 4y − 3 = 0.

• Group: (x² + 6x) + (y² − 4y) = 3.
• x: half of 6 is 3, squared is 9. y: half of −4 is −2, squared is 4. Add both to each side.
• (x² + 6x + 9) + (y² − 4y + 4) = 3 + 9 + 4 = 16.
• (x + 3)² + (y − 2)² = 16 → center (−3, 2), radius √16 = 4.

Worked Example 2 — only one variable needs work

Find the center and radius of x² + y² − 10x + 9 = 0.

• Group: (x² − 10x) + y² = −9.
• x: half of −10 is −5, squared is 25. Add 25 to both sides.
• (x − 5)² + y² = −9 + 25 = 16 → center (5, 0), radius 4.

(Oʻzbekcha: y allaqachon toʻliq kvadrat, faqat x bilan ishlaymiz.)

Worked Example 3 — read the radius carefully

After completing the square a circle becomes (x − 1)² + (y + 3)² = 20. State its center and radius.

• Center = (1, −3). Radius = √20 = 2√5 ≈ 4.47.

Trap: forgetting to add the new constants to the right side too. The equation must stay balanced, or the radius comes out wrong. (Oʻzbekcha: qoʻshilgan sonlarni oʻng tomonga ham qoʻshishni unutmang — muvozanat saqlanishi kerak.)

Practice 1

Find the center and radius of x² + y² + 8x + 2y + 8 = 0.

Practice 2

Find the center and radius of x² + y² − 6y = 0.

Key words — Kalit soʻzlar

• Complete the square — toʻliq kvadratga keltirish
• Standard form — standart koʻrinish
• Expanded form — yoyilgan koʻrinish
• Perfect square — toʻliq kvadrat
• Center — markaz
• Radius — radius
• Group (terms) — guruhlash (hadlar)
• Balance (the equation) — muvozanat (tenglama)
• Constant — oʻzgarmas (ozod had)

Summary

• Group x-terms and y-terms; complete the square on each using (half of the coefficient)².
• Add the new constants to both sides to keep balance.
• Rewrite as (x − h)² + (y − k)² = r², then read center (h, k) and radius √(right side).