Circles Class 10 ||Maths|| Chapter 10 NCERT Notes

Circles Class 10 ||Maths|| Chapter 10 NCERT Notes

1. Circle and Its Related Terms:

Before diving into tangents, it’s important to recap the basic definitions:

  • Circle: A circle is the set of all points in a plane that are at a fixed distance (called radius) from a fixed point (called the centre).
  • Radius: The distance between the centre and any point on the circle.
  • Chord: A line segment with its endpoints on the circle.
  • Diameter: The longest chord of a circle, which passes through the centre of the circle. The diameter is twice the radius.
  • Secant: A line that intersects the circle at two distinct points.
  • Tangent: A line that touches the circle at exactly one point.

2. Tangent to a Circle:

A tangent to a circle is a line that touches the circle at exactly one point, known as the point of contact.

Properties of a Tangent:

  • A tangent to a circle is perpendicular to the radius drawn to the point of contact.

    Theorem 1: The tangent at any point of a circle is perpendicular to the radius at the point of contact.

    Proof:

    • Let the point of contact be P and the centre of the circle be O.
    • The tangent at point P touches the circle at only that point.
    • If we assume a point Q on the tangent line different from P, then the line segment OQ will be longer than OP because OQ would intersect the circle at some other point (making it not a tangent anymore).
    • Hence, OP is the shortest distance between O and the tangent line, implying that OP is perpendicular to the tangent at P.

    Therefore, OP ⊥ tangent at P.

Important Points:
  • A circle can have infinitely many tangents, as a tangent can be drawn at any point on the circle's circumference.
  • From a point outside a circle, exactly two tangents can be drawn to the circle.

3. Length of Tangents from an External Point:

Another important concept in this chapter is the length of a tangent drawn from an external point to a circle. The length of the tangent from an external point to the point of contact on the circle is always equal for both tangents.

Theorem 2: The lengths of the two tangents drawn from an external point to a circle are equal.

Proof:

  • Let P be an external point from which two tangents PA and PB are drawn to the circle with centre O and point of contact A and B respectively.

  • We need to prove that PA = PB.

    Steps:

    1. Join OA and OB.
    2. In triangles OAP and OBP:
      • OA = OB (Radii of the circle).
      • ∠OAP = ∠OBP = 90° (Radius is perpendicular to the tangent).
      • OP = OP (Common side).

    By the RHS (Right angle-Hypotenuse-Side) congruence condition, triangles OAP and OBP are congruent.
    Therefore, PA = PB (corresponding parts of congruent triangles).

Corollary:

The two tangents drawn from an external point to a circle are equally inclined to the line joining the external point to the centre of the circle.

4. Tangent-Secant Theorem (Optional, Higher Concept):

This theorem states that if a secant and a tangent are drawn from a point outside the circle, then:

(Length of the tangent segment)² = (Length of external part of the secant) × (Total length of the secant).

This concept is more advanced and is not always tested in Class 10, but it's good to know for additional clarity.

5. Number of Tangents from Different Positions:

  1. Point on the Circle:
    Only one tangent can be drawn to the circle from a point on the circle itself.

  2. Point Inside the Circle:
    No tangent can be drawn from a point inside the circle.

  3. Point Outside the Circle:
    Exactly two tangents can be drawn from a point outside the circle.

6. Common Tangents to Two Circles:

There are cases where two circles may share common tangents. There are three types of common tangents:

  • Direct Common Tangents: Tangents that do not intersect the line segment joining the centres of the two circles.
  • Transverse Common Tangents: Tangents that intersect the line joining the centres of the circles.
  • No Tangent: If one circle lies inside the other, no tangent can be drawn between them.

Key Formulas:

  1. Length of Tangent from a point P to a circle with centre O and radius r:

    Length of Tangent=OP2−r2

    where OP is the distance from the external point P to the centre of the circle O.

  2. For two non-intersecting circles with centres O₁ and O₂ and radii r₁ and r₂, the length of the direct common tangent is given by:

    Length of direct common tangent=O1O22−(r1−r2)2

Important Theorems Recap:

  • Theorem 1: The tangent at any point of a circle is perpendicular to the radius at the point of contact.
  • Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.

Exercise Breakdown (NCERT):

Ex 10.1 and Ex 10.2 contain questions testing:

  • Understanding of tangents, secants, and their properties.
  • Application of theorems about tangents.
  • Calculating the lengths of tangents from external points.

That's a detailed breakdown of Chapter 10: Circles from Class 10 NCERT Maths. This chapter focuses on tangents and their properties, using logic and geometry to prove important theorems.

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