The limiting points of the coaxal system determined by the circles x^2+y^2-2 x-6 y+9=0 and x^2+y

The limiting points of the coaxal system determined by the...

Key Concepts

Coaxal (Coaxial) Circles

Coaxal circles refer to a family of circles that all share the same radical axis, meaning that the power of any point with respect to any circle in the family differs by a constant. These circles are related by a parameter, often forming what is known as a pencil of circles, and they either all pass through two fixed points or are defined such that every circle in the family is orthogonal to a given circle.

Limiting Points

In the context of coaxal circles, limiting points are the two fixed points that all circles of the family 'approach' or 'tend to' as a parameter in the circle equation varies. These points are common to all circles in the pencil and serve as the effective intersections of the family, even when it is not possible to define an actual intersection when parameters take extreme values.

Radical Axis

The radical axis of two circles is the locus of points where the power with respect to both circles is equal. It forms the common line from which all circles in a coaxal system derive their relative positions. The concept is fundamental in understanding how circles in a pencil are related, as every circle in the coaxal system shares this line as part of its geometric structure.

Transcript

00:01 Hello everyone, in this question two equations of circles are given.

00:05 Equation of circles.

00:08 First equation of circle is x square plus y square minus of 2x minus of 6y plus 9 is equal to 0.

00:18 And second equation of circle is x square plus y square plus 6x minus of 2y plus 1 is equal to 0.

00:28 And we have to find the limiting points of coaxial system of coaxal system of circles.

00:44 So first of all we find the radical axis of these two circles which is equal to s1 minus s2 is equal to zero.

00:56 So our first circle is x square plus y square minus of 2x minus of 6y plus 9 minus second circle x square minus y square minus 6x plus 2y minus 1 is equal to 0.

01:16 So x square positive of x square cancel out with negative similarly y square also cancel out.

01:22 When we add this, we get a minus of 8x and minus of 4y plus 8 is equal to 0.

01:32 Take minus 4 as common, we get 2x plus y minus of 2 is equal to 0.

01:42 So our radical axis become 2x plus y minus 2 is equal to 0.

01:48 Then our equation of circle, then our equation of any circle, any circle, any circle with given circle.

02:04 Coaxel with given circle becomes x square plus y square.

02:15 First circle x square plus y square minus of 2x, minus of 6y plus 9.

02:21 Is coaxial with second circle then its radical axis lambda times 2x plus y minus 2 is equals to 0 we can write this equation as x square plus y square minus 2 plus 2 times lambda so we get plus x multiply by 2 lambda minus 2 plus lambda minus 2 plus lambda minus 6 times and 9 minus 2 lambda is equal to 0.

02:58 From this equation, our center becomes center is equal to the center of the circle is 1 minus lambda and 1 minus lambda and 3 minus lambda divide by 2.

03:14 This is our circle.

03:16 Then it's radius.

03:17 Its radius is equal to square root of components of center square, 1 minus lambda whole square plus 3 minus lambda divide by 2 whole square minus minus minus its constant value which is 9 minus 2 lambda which end and we know in coaxial system in coexel system radius being 0.

03:53 Radius is equal to 0...

Please give Ace some feedback