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Revision Notes

Circles

Telangana Open School (TOSS) · Class 12 · Mathematics

Quick revision notes for Circles — Telangana Open School (TOSS) Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

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Key Topics to Revise

1

Standard and General Equation of a Circle

  • The standard form of the equation of a circle with centre (h, k) and radius r is: (x - h)² + (y - k)² = r².
  • The general form of the equation of a circle is: x² + y² + 2gx + 2fy + c = 0.
  • From the general form, the centre is (-g, -f) and the radius is √(g² + f² - c).
2

Special Cases and Diameter Form

  • If a circle passes through the origin, then c = 0 in the general form.
  • If the circle touches the x-axis, then |k| = r; if it touches the y-axis, then |h| = r.
  • When the endpoints of a diameter are (x₁, y₁) and (x₂, y₂), the equation of the circle is (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0.
3

Tangents, Normals, and Chords

  • The equation of the tangent to the circle S = 0 at point (x₁, y₁) is S₁ = 0.
  • The normal to a circle at a point passes through the centre and is perpendicular to the tangent.
  • The length of the tangent from an external point (x₁, y₁) to the circle S = 0 is √S₁₁.
4

Relative Position of Two Circles and Orthogonality

  • Two circles can intersect, touch externally/internally, or not touch at all depending on the distance between centres and sum/difference of radii.
  • Two circles are orthogonal if the angle between them is 90°, which happens when 2(gg' + ff') = c + c'.
  • The radical axis is the locus of points having equal power with respect to two circles and is given by S - S' = 0.

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Full Notes

Key Concepts

The standard form of the equationThe general secondIf the endpoints of a diameterThe tangent to a circle atTwo circles are said to be

Frequently Asked Questions

What are the important topics in Circles for Telangana Open School (TOSS) Class 12 Mathematics?
Circles covers several key topics that are frequently asked in Telangana Open School (TOSS) Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Circles — Telangana Open School (TOSS) Class 12 Mathematics?
Understand the core concepts first, then work through the 48 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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