SAT-79: Circle Equations in the Coordinate Plane

SAT-79: Circle Equations in the Coordinate Plane

Description: A circle on the coordinate plane has a tidy standard-form equation that hands you its center and radius directly. Reading them correctly — especially the signs — is the whole skill here.

The standard form

(x − h)² + (y − k)² = r²

• The center is (h, k).
• The radius is r = √(right-hand side).

Watch the signs: the form subtracts h and k, just like vertex form in SAT-35. (Oʻzbekcha: markaz (h, k); ishoralarga eʼtibor bering — formula h va k ni ayiradi.)

The sign trap

Because the equation has (x − h), a plus sign inside means a negative coordinate: (x + 3) is really (x − (−3)), so h = −3. And the right side is r², not r — you must take the square root. (Oʻzbekcha: (x + 3) aslida (x − (−3)), demak h = −3; oʻng tomon r² boʻlgani uchun ildiz oling.)

Worked Example 1 — read center and radius

Find the center and radius of (x − 2)² + (y − 5)² = 49.

• Center = (2, 5). Radius = √49 = 7.

Worked Example 2 — handle the signs

Find the center and radius of (x + 4)² + (y − 1)² = 25.

• (x + 4) → h = −4; (y − 1) → k = 1. Center = (−4, 1).
• Radius = √25 = 5.

(Oʻzbekcha: (x + 4) → h = −4, chunki formula ayiradi.)

Worked Example 3 — build the equation

Write the equation of a circle with center (3, −2) and radius 6.

• (x − 3)² + (y − (−2))² = 6².
• Equation: (x − 3)² + (y + 2)² = 36.

Tip: a negative center coordinate becomes a plus sign in the equation, and remember to square the radius on the right. (Oʻzbekcha: markazning manfiy koordinatasi tenglamada plyusga aylanadi; radiusni kvadratga koʻtaring.)

Worked Example 4 — build a circle from a center and a point

Write the equation of a circle centered at (1, 2) that passes through the point (4, 6).

• The radius is the distance from the center to that point: r = √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5.
• So r² = 25, giving (x − 1)² + (y − 2)² = 25.

This connects circles to the distance formula from SAT-70 — the radius is just a distance. (Oʻzbekcha: radius — markazdan nuqtagacha boʻlgan masofa, shuning uchun masofa formulasidan foydalanamiz.)

Is a point inside, on, or outside the circle?

Plug the point into the left side and compare with r². If it equals r², the point is on the circle; if it is less, the point is inside; if it is more, the point is outside. This is a quick SAT check that needs no graphing. (Oʻzbekcha: nuqtani tenglamaga qoʻyib, natijani r² bilan solishtiramiz.)

Practice 1

Find the center and radius of (x − 1)² + (y + 6)² = 16.

Practice 2

Write the equation of a circle with center (−5, 0) and radius 3.

Key words — Kalit soʻzlar

• Circle equation — aylana tenglamasi
• Standard form — standart koʻrinish
• Center — markaz
• Radius — radius
• Coordinate plane — koordinata tekisligi
• Square root — kvadrat ildiz
• Sign — ishora
• Coordinate (h, k) — koordinata (h, k)
• Squared — kvadratga koʻtarilgan

Summary

• Standard form: (x − h)² + (y − k)² = r².
• Center = (h, k); a plus sign inside means a negative coordinate.
• Radius = square root of the right side (it equals r², not r).