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Hello learners! 👋

we’re going to explore an essential topic in coordinate geometry – the Equation of a Circle. Whether you're just getting started with conic sections or preparing for an exam, this topic is foundational. Let's break it down step-by-step in a way that’s simple and easy to remember.

🧠 What is a Circle?

A circle is defined as the set of all points in a plane that are equidistant from a fixed point. This fixed point is called the center of the circle, and the distance from the center to any point on the circle is known as the radius.

Imagine drawing a circle on a graph. If you pick any point on the curve and measure its distance from the center, it will always be the same — that’s the magic of a circle!

What is a Circle

🧮 Equation of a Circle – Standard Form

The equation of a circle in standard form is:

(x - h)² + (y - k)² = r²

Equation of a Circle – Standard Form

Where:

  • (h, k) is the center of the circle
  • r is the radius

Example:
If a circle has a center at (3, 4) and a radius of 5, the equation becomes:

(x - 3)² + (y - 4)² = 25

🎯 Equation of a Circle – When Center is at the Origin

When the center of the circle is at the origin (0, 0), the equation becomes simpler:

x² + y² = r²

Equation of a Circle – When Center is at the Origin

Example:
If the radius is 7, then:

x² + y² = 49

This simplified form is useful in problems where the circle is centered at the origin.

🧾 The General Form of a Circle’s Equation

Sometimes, a circle is given in a more expanded form known as the general form:

x² + y² + 2gx + 2fy + c = 0

This version hides the center and radius, but you can find them by converting it to the standard form using a method called completing the square.

From this form:

  • Center: (-g, -f)
  • Radius: √(g² + f² - c)

✏️ Summary

  • A circle is a set of points at a constant distance from a fixed center.
  • The standard form of a circle is (x - h)² + (y - k)² = r².
  • If the center is at the origin, the equation becomes x² + y² = r².
  • The general form is x² + y² + 2gx + 2fy + c = 0 and can be simplified using algebra.

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Happy learning! ✨