Mathematics Class 11 - Circles Notes

Circles

Welcome to the chapter on Circles for Class 11. In this chapter, you will learn about the definition and properties of circles, equations of circles in different forms, tangents and normals, and solve problems involving circles. By the end of this chapter, you will be able to apply these concepts to geometry and coordinate geometry problems.

Key Concepts

• Circle: The set of all points in a plane that are at a fixed distance (radius) from a fixed point (center).
• Radius: The distance from the center to any point on the circle.
• Diameter: A chord passing through the center; it is twice the radius.
• Chord: A line segment joining any two points on the circle.
• Tangent: A line that touches the circle at exactly one point.
• Normal: A line perpendicular to the tangent at the point of contact.

Equation of a Circle

The standard form of the equation of a circle with center at (h, k) and radius r is:

(x - h)² + (y - k)² = r²

If the center is at the origin (0, 0), the equation becomes:

x² + y² = r²

General Equation of a Circle

The general form of a circle is:

x² + y² + 2gx + 2fy + c = 0

Here, the center is at (-g, -f) and the radius is √(g² + f² - c).

Tangent to a Circle

The equation of the tangent to the circle x² + y² = r² at the point (x₁, y₁) is:

x x₁ + y y₁ = r²

The length of the tangent from a point P(x₁, y₁) to the circle x² + y² + 2gx + 2fy + c = 0 is:

√[x₁² + y₁² + 2g x₁ + 2f y₁ + c]

Properties and Applications

• All radii of a circle are equal.
• The perpendicular from the center to a chord bisects the chord.
• Tangents from an external point to a circle are equal in length.
• The angle subtended by a diameter at the circumference is a right angle.

Practice Questions

1. Find the equation of a circle with center (2, -3) and radius 5.
2. Write the equation of a circle passing through the origin with center (4, 5).
3. Find the length of the tangent from the point (6, 8) to the circle x² + y² = 25.
4. What is the equation of the tangent to the circle x² + y² = 16 at the point (4, 0)?
5. If the equation of a circle is x² + y² - 4x + 6y - 12 = 0, find its center and radius.

Challenge Yourself

• Prove that the tangents drawn from an external point to a circle are equal in length.
• Find the equation of a circle passing through three given points.

Did You Know?

• The circle is the shape with the largest area for a given perimeter.
• The concept of π (pi) comes from the ratio of the circumference to the diameter of a circle.

Glossary

• Radius: Distance from the center to any point on the circle.
• Chord: A line segment joining two points on the circle.
• Tangent: A line that touches the circle at one point only.
• Normal: A line perpendicular to the tangent at the point of contact.

Answers to Practice Questions

1. (x - 2)² + (y + 3)² = 25
2. (x - 4)² + (y - 5)² = 41
3. Length = √[(6)² + (8)² - 25] = √[36 + 64 - 25] = √75 = 5√3
4. 4x = 16 ⇒ x = 4, so the tangent is x = 4 (or xx₁ + yy₁ = r²: 4x = 16)
5. Center: (2, -3); Radius: √(2² + (-3)² + 12) = √(4 + 9 + 12) = √25 = 5

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Practice problems on circles to master coordinate geometry and solve advanced questions!