Circles Class 10th Notes - Free NCERT Class 10 Maths Chapter 10 Notes

Q: Prove the tangents drawn at ends of a diameter of circle are parallel to each other.

Let there be a circle with a centre at O and diameter AB. At A, CD is the tangent to the circle and at B, EF is the tangent to the circle. We shall prove that CD EF. Now, OA ⟂ CD and OB ⟂ EF [radius of the circle is perpendicular to the tangent at the point of contact] Hence, between CD and EF, the alternate interior angles are equal (is of 90o). ⇒ CD EF [if the alternate angles are equal then lines are parallel].

Circles Class 10th Notes - Free NCERT Class 10 Maths Chapter 10 Notes - Download PDF

Edited By Ramraj Saini | Updated on Mar 19, 2022 12:51 PM IST

NCERT Class 10 Maths chapter 10 notes based on the circle, equation of circle graph of the circle, find radii of circle tangent and normal equation. The chapter circle Class 10 notes also includes how to draw a circle. CBSE Class 10 Maths chapter 10 notes include FAQ or frequently asked questions about the chapter. These topics can also be downloaded from Class 10 Maths chapter 10 notes pdf download.

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NCERT Class 10 Chapter 10 Notes-
Circle in Two Variables Class 10 Notes- Topic 1:
Definition of a Circle
Equation of a Circle with Centre (h, k) and Radius-
Equation of the Tangent:
Equation of Normal:
Equation of a Common Chord of the Two Circles-
If two circles with centres C1 and C2 and radii r and r’ respectively, then we have the following results regarding their common tangents.
Significance of NCERT Class 10 Maths Chapter 10 Notes
Class 10 Chapter Wise Notes
NCERT Solutions of Cass 10 Subject Wise
NCERT Class 10 Exemplar Solutions for Other Subjects:

Also, students can refer,

NCERT Class 10 Chapter 10 Notes-

Circle in Two Variables Class 10 Notes- Topic 1:

Definition of a Circle

A circle is the locus of a point in a plane so that its distance from a fixed point in the plane is constant. The fixed point is called the centre of the circle and the constant distance is called the radius

Equation of a Circle with Centre (h, k) and Radius-

Some results regarding circle

Position of a point with respect to a circle. Point P (x₁, y₁₎ lies outside, on, or inside a circle.

1644387515932

Here, r is the radius of the circle.

The point P is inside the circle when the distance is less than the radius. Then it is negative.

The point is outside the circle when the distance is greater than the radius. Then it is positive.

The point P is on the circle when distance and radius are equal. Then it is zero.

Equation of the tangent to the circle from a point P (x₁, y₁)

Equation of the Tangent:

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

xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0

It is the equation of the tangent.

An equation of the Normal to the circle x²+y²+2gx+2fy+c=0 from a point P (x₁, y₁)

Equation of Normal:

x²+y²+2gx+2fy+c=0

(y-y₁)/(y₁+f)=(x-x₁)/(x₁+g)

It is the equation of normal.

Equation of a Common Chord of the Two Circles-

S₁=x²+y²+2g₁x+2f₁y+c₁=0

S₂=x²+y²+2g₂x+2f₂y+c₂=0

S₁-S₂=0

If two circles with centres C₁ and C₂ and radii r and r’ respectively, then we have the following results regarding their common tangents.

(x-h)²+(y-k)²=r²

(x-h')²+(y-k')²=r'²

Case 1. When C₁ C₂> r+r’ i.e., the distance between the centers is greater than the sum of their radii, the two circles do not intersect with each other and 4 common tangents can be drawn to two circles.

Here C1and C2 are the centres of two circles.

In this case, two circles either touch each other externally or internally.

1644387516541

Case 2. When C₁ C₂ = r+r’ i.e., the distance between the centers is equal to the sum of the radii, the two circles touch each other externally, two direct common tangents are real and distinct and the transverse common tangents coincide.

1644387515605

Case 3. When C₁ C₂ < r+r’ i.e the distance between the centres is less than the sum of the radii, the circles intersect at two real and distinct points, the two direct common tangents are real and distinct while the transverse common tangents are imaginary.

1644387516235

Significance of NCERT Class 10 Maths Chapter 10 Notes

Circle Class 10 notes will be helpful to revise the chapter and to get an idea about the main topics covered in the chapter. CBSE Class 10 Maths chapter 10 notes also contain previous year’s questions and NCERT TextBook pdf. Circle Class 10 notes pdf download can be used to prepare in offline mode. Class 10 Maths chapter 10 notes contain frequently asked questions of the chapter.

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Frequently Asked Question (FAQs)

1. Prove the tangents drawn at ends of a diameter of circle are parallel to each other.

Let there be a circle with a centre at O and diameter AB. At A, CD is the tangent to the circle and at B, EF is

the tangent to the circle. We shall prove that CD EF. Now, OA ⟂ CD and OB ⟂ EF [radius of the circle is

perpendicular to the tangent at the point of contact] Hence, between CD and EF, the alternate interior

angles are equal (is of 90o). ⇒ CD EF [if the alternate angles are equal then lines are parallel].

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