THE CIRCLE | Listening

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THE CIRCLE THE CIRCLE

A circle is the locus of a point which moves, so that its distance from a fixed point, called the centre, is equal to a given distance.

The fixed point is called the centre of the circle and the constant distance is called its radius. i.e.,

One. Standard equation of a circle

If the centre of circle is at origin and radius is a, then the equation of circle is given by

X squared plus Y squared equals a squared and this is also known as standard equation of circle.

In case the centre is a point H, K, and not origin then the equation of circle is given by

X minus H squared plus Y minus K squared equals a squared C P equals radius equals a according to constraint defined by circle distance

C P equals radius equals a implies X minus H squared plus Y minus K squared equals a squared P X, Y C H, K. This form is known as central form of equation of circle.

Two. General Equation of a Circle:

Important remarks

Three. Other Forms of Circle

Three point two. Parametric form :

Four. Equation of a Circle in Special Cases

(ii) When the circle touches x-axis

(iii) When the circle touches y-axis.

(v) When the circle passes through the origin and centre lies on x-axis: Let centre of circle be

Illustration two

Illustration four

Illustration six

Second Method:

Find the equation of the circumcircle of the quadrilateral formed by the four lines ax plus by plus c equals zero and bx minus ay plus c equals zero

Find the shortest and largest distance from the point left parenthesis two, minus seven right parenthesis to the circle

Examine if the two circles x squared plus y squared minus two x minus four y equals zero and x squared plus y squared minus eight y minus four equals zero touch each other externally or internally.

Solution :

Case I:

Another Method:

Solution:

Eight. DIFFERENT FORMS OF THE EQUATION OF TANGENT

Eight point two. Parametric form:

Eight point three. Slope form:

Solution:

Second part:

Solution:

Five. Length of Tangent

Solution:

Ten. Equation of pair of tangents

Eleven. Director circle

x squared plus y squared equals two A squared Twelve. Normal to the Circle

T equals S one ,

Fifteen point two. Length of common chord :

Two. Let the equation to the circle be

Problems based on pole and polar with respect to a circle

Illustration thirty-three

Illustration thirty-four

Illustration thirty-five

Second part:

Seventeen. Family of Circles

Find the equation of a circle passing through the points (two, zero), (three, negative one) and (two, five). Solution :

Nineteen. Orthogonal Circles

Modified definition of radical axis:

Radical Centre:

Properties of Radical Centre:

Illustration forty-six

Given sides are

Twenty-one. Coaxial circles

Some properties

Twenty-two. The Common Tangents of Two circles

Case one: C two completely within C one

Case two: C two touches C one internally

Case three: C and C two intersect. not just touching.

Solution :

Solution:

SOME MORE PROBLEMS

Show that the circle circumscribing the triangle whose sides are

Overview

The document delves into the definition, equations, and various forms of circles in mathematics. It explains the relationship between the circle’s radius, center, and its equations while providing formulas to determine specific characteristics based on different conditions.

Key Points

1A circle is defined as the locus of points equidistant from a fixed center
2The standard equation of a circle is derived based on different coordinates and parameters
3The general equation of a circle can indicate real, imaginary, or point circles depending on certain conditions
4Various forms of circle equations exist, including parametric and diametric forms
5Finding the center and radius of a circle requires understanding its coefficients and relevant equations

Details

Category: Mathematics and Statistics

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